2. Intro duc ti on to Q uantum M ec hani c s
2.1 L a ws of Q uan tum Me c hanic s
2.2 State s , obs e rv able s and e ige n v alue s
2 .2 .1 P r o p er ties o f eig en fu n c tio n s
2 .2 .2 R eview o f lin ea r A lg eb r a
2.3 Me as ure me n t and probabilit y
2 .3 .1 W a v efu n c tio n c o lla p s e
2 .3 .2 P o s itio n mea s u r emen t
2 .3 .3 M o men tu m mea s u r emen t
2 .3 .4 E xp ec ta t io n v a lu es
2.4 E ne rgy e ige n v alue proble m
2.5 O p e rators , Comm utators and Unc e rtain t y Princ iple
2 .5 .2 C o mm u tin g o b s er v a b les
2 .5 .3 U n c er ta in t y p r in c ip le
2. 1 La w s of Quantum M ec hanic s
Ev er y p h y s ical th eor y is for m u lated in ter ms of math ematica l ob jects . It is th u s n eces s ar y to es tab lis h a s et of r u les to map p h y s ical con cep ts an d ob jects in to math ematical ob jects th at w e u s e to r ep r es en t th em. S ometimes th is map p in g is ev id en t, as in clas s ical mec h an ics , wh ile for oth er th eor ies , s u c h as q u an tu m mec h an ics , th e math ematical ob jects ar e n ot in tu itiv e.
In th e s ame w a y as clas s ical mec h an ics is fou n d ed on Newton ’s la ws or electr o d y n amics on th e M ax w ell-Boltzman n eq u ation s , q u an tu m mec h an ics is als o b as ed on s ome fu n d amen tal la ws , wh ic h ar e called th e p os tu lates or ax ioms of q u an tu m mec h an ics .
W e w an t in p ar ticu lar to d ev elop a math ematical mo d el for th e d y n amics of clos ed q u an tu m s y s tem s 1 : th er efor e w e ar e in ter es ted in d efi n in g
s tates – ob s er v ab les – meas u r emen ts – ev olu tion
S ome s u b tleties will ar is e s in ce w e ar e tr y in g to d efi n e meas u r emen t in a clos ed s y s tem, wh en th e meas u r in g p er s on is in s tead ou ts id e th e s y s tem its elf. A mor e comp lete p ictu r e, th at can ex p lain s ome of th e con fu s ion ar is in g fr om th e meas u r emen t p r o ces s , is p os s ib le, b u t w e will n ot s tu d y it in th is cou r s e.
W e ar e in ter es ted in giv in g a d es cr ip tion of p h y s ical p h en omen a an d in p ar ticu lar in h o w th ey emer ge d u r in g an ex p er imen t.
Ex p eri men ts – A p h y s ical ex p er imen t can b e d iv id ed in to t w o s tep s : p r ep ar at ion an d meas u r emen t. In clas s ical mec h an ics (CM ):
- th e fi r s t s tep d eter min es th e p os s ib le ou tcomes of th e ex p er imen t,
- wh ile th e meas u r emen t r etr iev es th e v alu e of th e ou tcome. In q u an tu m mec h an ics (Q M ) th e s itu ation is s ligh tly d iff er en t:
- th e fi r s t s tep (p r ep ar ation ) d eter min es th e pr ob abil ities of th e v ar iou s p os s ib le ou tcomes ,
- th e s econ d s tep (meas u r emen t) r etr iev e th e val u e of a p ar ticu lar ou tcome, in a s tatis tic man n er .
Th is s ep ar ation of th e ex p er imen t in t w o s tep s is r efl ected in to th e t w o t y p es of op er ator s th at w e fi n d in Q M .
- Th e fi r s t s tep cor r es p on d s to th e con cep t of a state of th e s y s tem,
1 W e d efi n e a clo s e d s ys t em a n y s ys t em th a t is is o la ted , th u s n o t exc h a n g in g a n y in p u t o r o u tp u t a n d n o t in ter a c tin g w ith a n y o th er s ys t em. A n o p en s ys t em in s t ea d in ter a c ts e.g ., w ith a n exter n a l en vir o n men t.
- wh ile th e s econ d s tep cor r es p on d s to obs erv ab les .
In CM th e s tate of a s y s tem is d es cr ib ed b y a s et of p r op er ties . F or ex amp le, if w e con s id er a b all, w e can d efi n e its s tate b y giv in g its p os ition , momen tu m, en er gy , an gu lar momen tu m (if for ex amp le th e b all is s p in n in g), its temp er atu r e etc. W e can th en p er for m a meas u r emen t on th is b a ll, for ex amp le meas u r in g its p os ition . Th is will giv e u s on e v alu e for on e p os s ib le ob s er v ab le (th e p os ition ).
{ }
W e can ex p r es s th is p r o ces s in math ematical ter ms . Th e s tate of th e s y s tem is d efi n ed b y a s et of v alu es : r r , p r , E , L r , T , . . . .
{ }
All of th es e v alu es (an d th er e migh t b e of cou r s e mor e th at I h a v en ’t wr itten d o wn ) ar e n eed ed to fu lly d es cr ib e th e s tate of th e b all. P er for min g a meas u r emen t of th e p os ition , will r etr iev e th e v alu es r x , r y , r z = r r (th e s ame v alu es th at d es cr ib e th e s tate).
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If w e n o w con s id er a n u cleu s , w e can as w ell giv e a d es cr ip tion of its s tate. In q u an tu m mec h an ics , a comp lete d es cr ip tion of th e s tate of a q u an tu m ob ject (or s y s tem) is giv en math ematically b y th e s tate v ector ψ (or w a v efu n ction ψ ( r r )). Th e s itu ation is h o w ev er d iff er en t th an in clas s ical mec h an ics .
Th e s tate v ector is n o lon ger a collection of v alu es for d iff er en t p r op er ties of th e s y s tem. Th e s tate giv es in s tead a comp lete d es cr ip tion of th e s et of pr ob abil ities for all th e p h y s ical p r op er ties (or ob s er v ab les ). All th e in for mation is con tain ed in th e s tate, ir r es p ectiv ely on h o w I got th e s tate , of its p r ev iou s h is tor y .
O n th e oth er h an d , th e ob s er v ab les ar e all th e p h y s ical p r op er ties th at in p r in cip le can b e meas u r ed , in th e s ame w a y as it w as in clas s ical mec h an ics . S in ce h o w ev er th e s tate on ly giv es p r ob ab ilities for all ob s er v ab les , th e r es u lt of meas u r emen t will b e a s tatis tical v ar iab le.
All of th es e con s id er ation s ar e mad e mor e for mal in th e ax ioms of q u an tu m mec h an ics th at als o in d icate th e math ematical for malis m to b e u s ed .
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1. Th e p r op er ties of a q u an tu m s y s tem ar e comp letely d efi n ed b y s p ecifi cation of its s tate v ector ψ . Th e s tate v ector is an elemen t of a comp lex Hilb er t s p ace H called th e s p ace of s tates .
A
2. W ith ev er y p h y s ical p r op er t y (en er gy , p os ition , momen tu m, an gu lar momen tu m, ...) th er e ex is ts an as s o ciated lin ear , Her mitian op er ator A (u s u ally called ob s er v ab le), wh ic h acts in th e s p ace of s tates H . Th e eigen v alu es of th e op er ator ar e th e p os s ib le v alu es of th e p h y s ical p r op er ties .
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| ) | )
3. a If ψ is th e v ector r ep r es en tin g th e s tate of a s y s tem an d if ϕ r ep r es en ts an oth er p h y s ical s tate, th er e ex is ts a p r ob ab ilit y p ( ψ , ϕ ) of fi n d in g ψ in s tate ϕ , wh ic h is giv en b y th e s q u ar ed mo d u lu s of th e in n er p r o d u ct on
H | ) | ) |( | )|
: p ( ψ , ϕ ) = ψ ϕ 2 (Bor n Ru le).
|( | )|
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3. b If A is an ob s er v ab le with eigen v alu es a n an d eigen v ector s n [s u c h th at th e eigen v alu e eq u ation is A n = a n n ], giv en a s y s tem in th e s tate ψ , th e p r ob ab ilit y of ob tain in g a n as th e ou tcome of th e meas u r emen t of A is p ( a n ) = n ψ 2 . After th e meas u r emen t th e s y s tem is left in th e s tate p r o jected on th e s u b s p ace of th e eigen v alu e a n (W a v e fu n ction collap s e).
4. Th e ev olu tion of a clos ed s y s tem is u n itar y (r ev er s ib le). Th e ev olu tion is giv en b y th e time-d ep en d en t S c h r ¨ od in ger eq u ation
∂ t
i i ∂ | ψ ) = H | ψ )
H
wh er e is th e Hamilton ian of th e s y s tem (th e en er gy op er ator ) an d i is th e r ed u ced Plan c k con s tan t h/ 2 π (with
h th e Plan c k con s tan t, allo win g con v er s ion fr om en er gy to fr eq u en cy u n its ).
2. 2 States , obs ervables and eigenvalues
D | )
: F r om th e fi r s t p os tu late w e s ee th at th e s tate of a q u an tu m s y s te m is giv en b y th e s tate v ector ψ ( t ) (or th e w a v efu n ction ψ ( r x , t )). Th e s tate v ector con tain s all p os s ib le in for mation ab ou t th e s y s tem. Th e s tate v ector is a v ector in th e Hilb er t s p ace. A Hilb er t s p ace H is a comp lex v ector s p ace th at p os s es s an in n er p r o d u ct.
An ex amp le of Hilb er t s p ace is th e u s u al Eu clid ean s p ace of geometr ic v ector s . Th is is a p ar ticu lar ly s imp le cas e s in ce th e s p ace in th is cas e is r eal. In gen er al as w e will s ee, Hilb er t s p ace v ector s can b e comp lex (th at is , s ome of th eir comp on en ts can b e comp lex n u m b er s ). In th e 3D Eu clid ean s p ace w e can d efi n e v ector s , with a r ep r es en tation s u c h as r v = { v x , v y , v z } or :
v x
r v = v y v z
{ }
Th is r ep r es en tation cor r es p on d s to c h o os e a p ar ticu lar b as is for th e v ector (in th is cas e, th e u s u al x, y , z co or d i n ates ). W e can als o d efi n e th e in n er p r o d u ct b et w een t w o v ector s , r v an d r u (wh ic h is ju s t th e u s u al s calar p r o d u ct):
u x
r v · r u = [ v x v y v z ] · u y = v x u x + v y u x + v z u z
u z
Notice th at w e h a v e tak en th e tr an s p os e of th e v ector r v , r v T in or d er to calcu late th e in n er p r o d u ct. In gen er al, in a Hilb er t s p ace, w e can d efi n e th e d u al of an y v ector . Th e D ir ac n otation mak es th is mor e clear .
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(· | ( |
| ) | ·)
Th e n otation ψ is called th e D ir ac n otation an d th e s y m b ol is called ket . Th is is u s efu l in calcu latin g in n er p r o d u cts of s tate v ector s u s in g th e br a (wh ic h is th e d u al of th e k et), for ex amp le ϕ . An in n er p r o d u ct is th en wr itten as ϕ ψ (th is is a br acket , h en ce th e n ames ).
( | )
W e will often d es cr ib e s tates b y th eir w a v efu n ction in s tead of s tate v ector . Th e w a v efu n ction is ju s t a p ar ticu lar w a y of wr itin g d o wn th e s tate v ector , wh er e w e ex p r es s th e s tate v ector in a b as is lin k ed to th e p os ition of a p ar ticle its elf (th is is called th e p os ition r ep r es en tation ). Th is p ar ticu lar cas e is h o w ev er th e on e w e ar e mos tly in ter es ted in th is cou r s e. M ath ematically , th e w a v efu n ction is a comp lex fu n ction of s p ace an d time. In th e p os ition r ep r es en tation (th at is , th e p os ition b as is ) th e s tate is ex p r es s ed b y th e w a v efu n ction v ia th e in n er p r o d u ct ψ ( x ) = x ψ .
Th e p r op er ties of Hilb er t s p aces , k ets an d b r as an d of th e w a v efu n ction can b e ex p r es s ed in a mor e r igor ou s math e matical w a y . In th is cou r s e as s aid w e ar e mos tly in ter es ted in s y s tems th at ar e n icely d es cr ib ed b y th e w a v efu n ction . Th u s w e will ju s t u s e th is math ematical to ol, with ou t d elv in g in to th e math ematical d etails . W e will s ee s ome mor e p r op er ties of th e w a v efu n ction on ce w e h a v e d efi n ed ob s er v ab les an d meas u r emen t.
D
: All p h y s ical ob s er v ab les (d efi n ed b y th e p r es cr ip tion of ex p er imen t or meas u r emen t ) ar e r ep r es en ted b y a lin ear op er ator th at op er ates in th e Hilb er t s p ace H (a lin ear , comp lex , in n er p r o d u ct v ector s p ace).
∈
→ ∈
In math ematics , an op er ator is a t y p e of fu n ction th at acts on fu n ction s to p r o d u ce oth er fu n ction s . F or mally , an op er ator is a map p in g b et w een t w o fu n ction s p aces 2 A : g ( I ) f ( I ) th at as s ign s to eac h fu n ction g g ( I ) a fu n ction f = A ( g ) f ( I ).
H
Ex amp les of ob s er v ab les ar e wh at w e alr ead y men tion ed , e.g. p os ition , momen tu m, en er gy , an gu lar momen tu m. Th es e op er ator s ar e as s o ciated to clas s ical v ar iab les . T o d is tin gu is h th em fr om th eir clas s ical v ar iab le cou n ter p ar t, w e will th u s p u t a hat on th e op er ator n ame. F or ex amp le, th e p os ition op er ator s will b e x ˆ , y ˆ , z ˆ . Th e momen tu m op er ator s p ˆ x , p ˆ y , p ˆ z an d th e an gu lar momen tu m op er ator s L ˆ x , L ˆ y , L ˆ z . Th e en er gy op er ator is called Hamilton ian (th is is als o tr u e in clas s ical mec h an ics ) an d is u s u ally d en oted b y th e s y m b ol .
Th er e ar e als o s ome op er ator s th at d o n ot h a v e a clas s ical cou n ter p ar t (r emem b er th at q u an tu m-mec h an ics is mor e gen er al th an clas s ical mec h an ics ). Th is is th e cas e of th e s p in op er ator , an ob s er v ab le th at is as s o ciated to eac h p ar ticle (electr on , n u cleon , atom etc.). F or ex amp le, th e s p in of an electr on is u s u ally d en oted b y S ; th is is als o a v ector v ar iab le (i.e. w e can d efi n e S x , S y , S z ). I am omittin g h er e th e hat s in ce th er e is n o clas s ical v ar iab le w e can con fu s e th e s p in with . W h ile th e p os ition , momen tu m etc. ob s er v ab le ar e con tin u ou s op er ator , th e s p in is a d is cr ete op er ator .
Th e s econ d p os tu late s tates th at th e p os s ib le v alu es of th e p h y s ical p r op er ties ar e giv en b y th e eigen v alu es of th e op er ator s .
D : Ei genval ues and ei genfunc t i ons of an op er ator ar e d efi n ed as th e s olu tion s of th e eigen v alu e p r ob lem:
A [ u n ( r x )] = a n u n ( r x )
wh er e n = 1 , 2 , . . . in d ex es th e p os s ib le s olu tion s . Th e a n ar e th e eigen v alu es of A (th ey ar e s calar s ) an d u n ( r x ) ar e th e eigen fu n ction s .
·
Th e eigen v alu e p r ob lem con s is ts in fi n d in g th e fu n ction s s u c h th at wh en th e op er ator A is ap p lied to th em, th e r es u lt is th e fu n ction its elf m u ltip lied b y a s calar . (Notice th at w e in d icate th e action of an op er ator on a fu n ction b y A [ f ( )]).
y
2
i 0
z
2
0 − 1
Y ou s h ou ld h a v e s een th e eigen v alu e p r ob lem in lin ear algeb r a, wh er e y ou s tu d ied eigen v ector s an d eigen v alu es of matr ices . Con s id er for ex amp le th e s p in op er ator for th e electr on S . Th e s p in op er ator can b e r ep r es en ted b y th e follo win g matr ices (th is is called a matr ix r ep r es en tation of th e op er ator ; it’s n ot u n iq u e an d d ep en d s on th e b as is c h os en ):
x
2
1 0
S = 1 0 1 , S
= 1 0 − i , S
= 1 1 0
2
W e can calcu late wh at ar e th e eigen v alu es an d eigen v ector s of th is op er ator s with s ome s imp le algeb r a. In clas s w e con s id er ed th e eigen v alu e eq u ation s for S x an d S z . Th e eigen v alu e p r ob lem can b e s olv ed b y s ettin g th e d eter min an t of th e matr ix S α − s 1 1 eq u al to zer o. W e fi n d th at th e eigen v alu es ar e ± 1 for b oth op er ator s . Th e eigen v ector s ar e
d iff er en t:
v z = 1 , v z = 0
1 0
2 1
v 1 = √ 2
,
v 2 = √ 2 − 1
x 1 1 x 1 1
1
2 A fu n c tio n s p a c e f ( I ) is a c o llec tio n o f fu n c tio n s s a tis fyin g c er ta in p r o p er ties.
·
W e p r o v ed als o th at v 1 v 2 = 0 (th at is , th e eigen v ector s ar e or th ogon al) an d th at th ey for m a comp lete b as is (w e can wr ite an y oth er v ector , d es cr ib in g th e s tate of th e electr on s p in , as a lin ear com b in ation of eith er th e eigen v ector s of S z or of S x ).
d x
Th e eigen v alu e p r ob lem can b e s olv ed in a s imilar w a y for con tin u ou s op er ator s . Con s id er for ex amp le th e d iff er en tial op er ator , d [ · ] . Th e eigen v alu e eq u ation for th is op er ator r ead s :
d f ( x )
d x
wh er e a is th e eigen v alu e an d f ( x ) is th e eigen fu n ction .
= af ( x )
? Qu estion : wh at is f ( x )? W h at ar e all th e p os s ib le eigen v alu es (an d th eir cor r es p on d in g eigen fu n ction s )?
d x
Ex amp l es – Th e eigen v alu e eq u ation for th e op er ator is x d [ · ] is :
wh ic h is s olv ed b y f ( x ) = x n , a = n .
d f ( x )
x = af ( x )
d x
√ −
Th e ”s tan d ar d ” G au s s ian fu n ction 1 e x 2 / 2 is th e eigen fu n ction of th e F ou r ier tr an s for m. Th e F ou r ier tr an s for m is
2 π
an op er ation th at tr an s for ms on e comp lex -v alu ed fu n ction of a r eal v ar iab le in to an oth er on e (th u s it is an op er ator ):
F x : f ( x ) → f ( k ) , with f ( k ) = F x [ f ( x )]( k ) = √ 2 π
f ( x ) e
dx
˜ ˜ 1 J ∞ − ik x
− ∞
: f ( k ) → f ( x ) , f ( x ) = √ 2 π
f ( k ) e
dk
Notice th at s ometimes d iff er en t n or malization s ar e u s ed . W ith th is d efi n ition , w e als o fi n d th at th e in v er s e F ou r ier tr an s for m is giv en b y :
F k
− 1 ˜
1 J ∞ ˜
ik x
− ∞
Let’s n o w tu r n to q u an tu m mec h an ical op er ator s .
D
: P os i t i on op erat o r – Th e p os ition op er ator for a s in gle p ar ticle r x ˆ is s imp ly giv en b y th e s calar r x . Th is mean s th at th e op er ator r x ˆ actin g on th e w a v efu n ction ψ ( r x ) s imp ly m u ltip lies th e w a v efu n ction b y r x . W e can wr ite
r x ˆ [ ψ ( r x )] = r x ψ ( r x ) .
W e can n o w con s id er th e eigen v alu e p r ob lem for th e p os ition op er ator . F or ex amp le, for th e x -comp on en t of r x th is is wr itten as :
x ˆ [ u n ( x )] = x n u n ( x ) → x u n ( x ) = x n u n ( x )
wh er e w e u s ed th e d efi n ition of th e p os ition op er ator . Her e x n is th e eigen v alu e an d u n ( x ) th e eigen fu n ction . Th e s olu tion to th is eq u ation is n ot a p r op er fu n ction , b u t a d is tr ib u tion (a gen er alized fu n ction ): th e D ir ac d elta fu n ction : u n ( x ) = δ ( x − x n )
− ∞
fu n ction als o h as th e p r op er t y th at
∞ δ ( x ) dx = 1 an d of cou r s e x δ ( x − x 0 ) = x 0 δ ( x − x 0 ) (wh ic h cor r es p on d s to
D : Di rac Del t a func t i on δ ( x − x 0 ) J is eq u al to zer o ev er y wh er e ex cep t at x 0 wh er e it is in fi n ite. Th e D ir ac D elta
th e eigen v alu e p r ob lem ab o v e). W e als o h a v e:
J dx δ ( x − x 0 ) f ( x ) = f ( x 0 )
Th at is , th e in tegr al of an y fu n ction m u ltip lied b y th e d elta fu n ction giv es b ac k th e fu n ction its elf ev alu ated at th e p oin t x 0 . [S ee an y tex tb o ok (an d r ecitation s ) for oth er p r op er ties .]
Ho w man y s olu tion s ar e th er e to th e eigen v alu e p r ob lem d efi n ed ab o v e for th e p os ition op er ator ? O n e p er eac h p os s ib le p os ition , th at is an in fi n ite n u m b er of s olu tion s . Con v er s ely , all p os s ib le p os ition s ar e allo w ed v alu es for th e meas u r emen t of th e p os ition (a con tin u u m of s olu tion s in th is cas e).
D
: M oment um op erat o r – Th e momen tu m op er ator is d efi n ed (in an alogy with clas s ical mec h an ics ) as th e gen er ator of tr an s lation s . Th is mean s th at th e momen tu m mo d ifi es th e p os ition of a p ar ticle fr om r x to r x + d r x . It is p os s ib le to s h o w th at th is d efi n ition giv es th e follo win g for m of th e p os ition op er ator (in th e p os ition r ep r es en tation , or p os ition b as is )
p ˆ x = − i i ∂ x , p ˆ y = − i i ∂ y , p ˆ z = − i i ∂ z
∂
∂
∂
f(x)
δ(x - x 0 )
ε → 0
x 0 -ε/2
x 0 +ε/2
f(x)
δ(x - x 0 )
1/ε
x 0 x x 0 x
F ig . 9 : S c h ema tic s o f Dir a c ’s d elta fu n c tio n . Left: th e r ec t a n g u la r fu n c tio n o f b a s e ǫ a n d h eig h t ǫ b ec o mes th e d elta -fu n c tio n (r ig h t) in th e limit o f ǫ → 0 .
or in v ector n otation p ˆ = − i i ∇ . Her e i is th e r ed u ced Plan c k con s tan t h/ 2 π (with h th e Plan c k con s tan t) with v alu e
i = 1 . 054 × 10 − 34 J s .
Plan c k ’s con s tan t is in tr o d u ced in or d er to mak e th e v alu es of q u an tu m ob s er v ab les con s is ten t with th e cor r es p on d in g clas s ical v alu es .
W e n o w s tu d y th e momen tu m op er ator eigen v alu e p r ob lem in 1D . Th e p r ob lem’s s tatemen t is
p ˆ x [ u n
( x )] = p n
u ( x ) i i ∂ u n ( x ) = p u
→ −
n ∂ x n n
( x )
Th is is a d iff er en tial eq u ation th at w e can s olv e q u ite eas ily . W e s et k = p/ i an d call k th e w a v en u m b er (for r eas on s clear in a momen t). Th e d iff er en tial eq u ation is th en
wh ic h h as as s olu tion th e comp lex fu n ction :
∂ u n ( x ) = ik u
∂ x n n
( x )
k
n
u ( x ) = Ae ik n x = Ae i p n x
Th e momen tu m eigen fu n ction s an d eigen v alu es ar e th u s u n = Ae ik n x an d k n .
No w r emem b er th e mean in g of th e eigen v alu es . By th e s econ d p os tu late, th e eigen v alu es of an op er ator ar e th e p os s ib le v alu es th at on e can ob tain in a meas u r emen t.
O b s . 1 Th er e ar e n o r es tr iction s on th e p os s ib le v alu es ob tain ed fr o m a momen tu m meas u r emen ts . All v alu es p = i k
ar e p os s ib le.
O b s . 2 Th e eigen fu n ction u n ( x ) cor r es p on d s to a w a v e tr a v elin g to th e r igh t with momen tu m p n = i k n . Th is w as als o ex p r es s ed b y D e Br oglie wh en h e p os tu lated th e ex is ten ce of matter w a v es .
Lou is d e Br oglie (1892-1987) w as a F r en c h p h y s icis t. In h is Ph .D th es is h e p os tu lated a r elation s h ip b et w een th e momen tu m of a p ar ticle an d th e w a v elen gth of th e w a v e as s o cia ted with th e p ar ticle (1922). In d e Br oglie’s eq u ation a p ar ticle w a v elen gth is th e Plan c k ’s con s tan t d iv id ed b y th e p ar ticle momen tu m. W e can s ee th is b eh a v ior in th e electr on in ter fer ometer v id eo 3 . F or clas s ical ob jects th e momen tu m is v er y lar ge (s in ce th e m as s is lar ge), th en th e w a v elen gth is v er y s mall an d th e ob ject lo os e its w a v e b eh a v ior . D e Br oglie eq u ation w as ex p er imen tally con fi r med in 1927 wh en p h y s icis ts Les ter G er mer an d Clin ton D a v is s on fi r ed electr on s at a cr y s tallin e n ic k el tar get an d th e r es u ltin g d iff r action p atter n w as fou n d to matc h th e p r ed icted v alu es .
2.2.1 P rop erti es of ei genfunc ti ons
F r om th es e ex amp les w e can n otice t w o p r op er ties of eigen fu n ction s wh ic h ar e v alid for an y op er ator :
3 A . T o n o m u r a , J. E n d o , T . M a ts u d a , T . Ka w a s a ki a n d H . E z a w a , A m. J. o f P h ys . 5 7 , 1 1 7 (1 9 8 9 )
( | ) J
1. Th e eigen fu n ction s of an op er ator ar e or th ogon al fu n ction s . W e will as w ell as s u me th at th ey ar e n or malized . Con s id er t w o eigen fu n ction s u n , u m of an op er ator A an d th e in n er p r o d u ct d efi n ed b y f g = d 3 x f ∗ ( x ) g ( x ). Th en w e h a v e
J d 3 x u ∗ m ( x ) u n ( x ) = δ nm
2. Th e s et of eigen fu n ction s for ms a comp l ete b asi s .
Th is mean s th at an y oth er fu n ction can b e wr itten in ter ms of th e s et of eigen fu n ction s { u n ( x ) } of an op er ator A :
f ( x ) = Σ c n u n ( x ) , with c n = J d 3 x u ∗ n ( x ) f ( x )
n
[Note th at th e las t eq u alit y is v alid iif th e eigen fu n ction s ar e n or malized , wh ic h is ex actly th e r eas on for n or malizin g th em].
If th e eigen v alu es ar e a con tin u ou s p ar ameter , w e h a v e a con tin u u m of eigen fu n ction s , an d w e will h a v e to r ep lace th e s u m o v er n with an in tegr al.
Con s id er th e t w o ex amp les w e s a w. F r om th e p r op er t y of th e D ir ac D elta fu n ction w e k n o w th at w e can wr ite an y fu n ction as :
f ( x ) = J dx ′ δ ( x ′ − x ) f ( x ′ )
—
W e can in ter p r et th is eq u ation as to s a y th at an y fu n ction can b e wr itten in ter ms of th e p os ition eigen fu n ction δ ( x ′ x ) (n otice th at w e ar e in th e con tin u ou s cas e men tion ed b efor e, s in ce th e x -eigen v alu e is a con tin u ou s fu n ction ). In th is cas e th e co efficien t c n b ecomes als o a con tin u ou s fu n ction
c n → c ( x n ) = J
dx δ ( x − x n ) f ( x ) = f ( x n ) .
Th is is n ot s u r p r is in g as w e ar e alr ead y ex p r es s in g ev er y th in g in th e p os ition b as is .
If w e w an t in s tead to ex p r es s th e fu n ction f ( x ) u s in g th e b as is giv en b y th e momen tu m op er ator eigen fu n ction s w e h a v e: (con s id er 1D cas e)
f ( x ) = J dk u k ( x ) c ( k ) = J dk e ik x c ( k )
wh er e again w e n eed an in tegr al s in ce th er e is a con tin u u m of p os s ib le eigen v alu es . Th e co efficien t c ( k ) can b e calcu lated fr om
c ( k ) = J dx u ∗ k ( x ) f ( x ) = J dx e − ik x f ( x )
W e th en h a v e th at c ( k ) is ju s t th e F ou r ier tr an s for m of th e fu n ction f ( x ) (u p to a m u ltip lier ).
Th e F ou r ier tr an s for m is an op er ation th at tr an s for ms on e comp lex -v alu ed fu n ction of a r eal v ar iab le in to an oth er :
F x : f ( x ) → f ( k ) , with f ( k ) = F x [ f ( x )]( k ) = √ 2 π
f ( x ) e
dx
˜ ˜ 1 J ∞ − ik x
− ∞
: f ( k ) → f ( x ) , f ( x ) = √ 2 π
− ∞
f ( k ) e
dk
Notice th at s ometimes d iff er en t n or malization s ar e u s ed . W ith th is d efi n ition , w e als o fi n d th at th e in v er s e F ou r ier tr an s for m is giv en b y :
F k
− 1 ˜
1 J ∞ ˜
ik x
2.2.2 R evi ew of l i nea r A l geb ra
Th is is a v er y con cis e r ev iew of con cep ts in lin ear algeb r a, r ein tr o d u cin g s ome of th e id eas w e s a w in th e p r ev iou s p ar agr ap h s in a s ligh tly mor e for mal w a y .
A . V ecto rs a n d vecto r sp a ces
Q u an tu m mec h an ics is a lin ear th eor y , th u s it is w ell d es cr ib ed b y v ector s an d v ector s p aces . V ector s ar e math ematical ob jects (d is tin ct fr om s calar s ) th at can b e ad d ed on e to an oth er an d m u ltip lied b y a s calar . In Q M w e d en ote v ector s b y th e D ir ac n otation : | ψ ) , | ϕ ) , . . . Th en , th es e h a v e th e p r op er ties :
– If | ψ 1 ) an d | ψ 2 ) ar e v ector s , th en | ψ 3 ) = | ψ 1 ) + | ψ 2 ) is als o a v ector .
– G iv en a s calar s , | ψ 4 ) = s | ψ 1 ) is als o a v ector .
A v ector s p ace is a collection of v ector s . F or ex amp le, for v ector s of fi n ite d imen s ion s , w e can d efi n e a v ector s p ace of d imen s ion s N o v er th e comp lex n u m b er s as th e collection of all comp lex -v alu ed N-d imen s ion al v ector s .
· · ·
Ex amp le 1 – A familiar ex amp le of v ector s an d v ector s p ace ar e th e Eu clid e an v ector s an d th e r eal 3D s p ace. Ex amp le 2 – An oth er ex amp le of a v ector s p ace is th e s p ace of p oly n omials o f or d er n . Its elemen ts , th e p oly n omials P n = a 0 + a 1 x + a 2 x 2 + + a n x n can b e p r o v ed to b e v ector s s in ce th ey can b e s u mmed to ob tain an oth er p oly n omial an d m u ltip lied b y a s calar . Th e d imen s ion of th is v ector s p ace is n + 1.
∀
Ex amp le 3 – In gen er al, fu n ction s can b e con s id er ed v ector s of a v ector s pace with in fi n ite d imen s ion (of cou r s e, if w e r es tr ict th e s et of fu n ction s th at b elon g to a giv en s p ace, w e m u s t en s u r e th at th is is s till a w ell-d efi n ed v ector s p ace. F or ex amp le, th e collection of all fu n ction f ( x ) b ou n d ed b y 3 [ f ( x ) < 3 , x ] is n ot a w ell d efi n ed v ector -s p ace, s in ce s f ( x ) (with s a s calar > 1) is n ot a v ector in th e s p ace.
B . In n er p ro d u ct
( | ) | ) | )
W e d en ote b y ψ ϕ th e s calar p r o d u ct b et w een th e t w o v ector s ψ an d ϕ . Th e in n er p r o d u ct or s calar p r o d u ct is a map p in g fr om t w o v ector s to a comp lex s calar , with th e follo win g p r op er ties :
– It is lin ear in th e s econ d ar gu men t: ( ψ | a 1 ϕ 1 + a 2 ϕ 2 ) = a 1 ( ψ | ϕ 1 ) + a 2 ( ψ | ϕ 2 ) .
– It h as th e p r op er t y of comp lex con ju gation : ( ψ | ϕ ) = ( ϕ | ψ ) ∗ .
– It is p os itiv e-d efi n ite : ( ψ | ψ ) = 0 ⇔ | ψ ) = 0.
∫
· | || |
Ex amp le 1 – F or Eu clid ean v ector s th e in n er p r o d u ct is th e u s u al s calar p r o d u ct r v 1 r v 2 = r v 1 r v 2 cos ϑ . Ex amp le 2 – F or fu n ction s , th e in n er p r o d u ct is d efi n ed as :
( f | g ) = ∞ f ( x ) ∗ g ( x ) dx
− ∞
C. L in ea rly in d ep en d en t vecto rs (a n d fu n ction s)
| ) | ) | )
W e can d efi n e lin ear com b in ation s of v ector s as ψ = a 1 ϕ 1 + a 2 ϕ 2 + If a v ector can n ot b e ex p r es s ed as a
Σ
lin ear s u p er p os ition of a s et of v ector s , th an it is s aid to b e lin ear ly in d ep en d en t fr om th es e v ector s . In math ematical ter ms , if
| ξ ) / = a i | ϕ i ) ,
i
∀ a i
th en | ξ ) is lin ear ly in d ep en d en t of th e v ector s {| ϕ i )} .
D. B a sis
A b as is is a lin ear ly in d ep en d en t s et of v ector s th at s p an s th e s p ace. Th e n u m b er of v ector s in th e b as is is th e v ector s p ace d imen s ion . An y oth er v ector can b e ex p r es s ed as a lin ear com b in ation of th e b as is v ector s . Th e b as is is n ot u n iq u e, an d w e will u s u ally c h o os e an or th on or mal b as is .
{ }
Ex amp les – F or th e P oly n omial v ector s p ace, a b as is ar e th e mon omials x k , k = 0 , . . . , n . F or Eu clid ean v ector s th e v ector s alon g th e 3 co or d in ate ax es for m a b as is .
W e h a v e s een in clas s th at eigen v ector s of op er ator s for m a b as is .
E. Un ita ry a n d Hermitia n op era to rs
An imp or tan t clas s of op er ator s ar e s elf ad join t or Her mitian op er ator s , as ob s er v ab les ar e d es cr ib ed b y th em. W e n eed fi r s t to d efi n e th e ad join t of an op er ator A . Th is is d en oted A † an d it is d efi n ed b y th e r elation :
( A † ψ ) ϕ ) = ( φ | ( Aψ ) ) ∀ {| ψ ) , | ϕ )} .
Th is con d ition can als o b e wr itten (b y u s in g th e s econ d p r op er t y of th e in n er p r o d u ct) as :
( ψ | A † | ϕ ) = ( ϕ | A | ψ ) ∗
If th e op er ator is r ep r es en ted b y a matr ix , th e ad join t of an op er ator is th e con ju gate tr an s p os e of th at op er ator :
A † k ,j = ( k | A † | j ) = ( j | A | k ) ∗ = A j ∗ ,k .
D : Sel f-adjoi nt . A s elf ad join t op er ator is an op er ator s u c h th at A † = A , or mor e p r ecis ely
( ψ | A | ϕ ) = ( ϕ | A | ψ ) ∗
F or matr ix op er ator s , A k i = A ∗ ik .
An imp or tan t p r op er ties of Her mitian op er ator s is th at th eir eigen v alu es ar e alw a y s r eal (ev en if th e op er ator s ar e
d efi n ed on th e comp lex n u m b er s ). Th en , all th e ob s er v ab les m u s t b e r ep r es en ted b y h er mitian op er ator s , s in ce w e w an t th eir eigen v alu es to b e r eal, as th e eigen v alu es ar e n oth in g els e th an p os s ib le ou tcomes of ex p er imen ts (an d w e w ou ld n ’t w an t th e p os ition of a p ar ticle, for ex amp le, to b e a comp lex n u m b er ).
Th en , for ex amp le, th e Hamilton ian of an y s y s tem is an h er mitian op er ator . F or a p ar ticle in a p oten tial, it’s eas y to c h ec k th at th e op er ator is r eal, th u s it is als o h er mitian .
D : U ni t a ry op erat o rs U ar e s u c h th at th eir in v er s e is eq u al to th eir ad join t: U − 1 = U † , or
U U † = U † U = 1 1 .
W e will s ee th at th e ev olu tion of a s y s tem is giv en b y a u n itar y op er ator , wh ic h imp lies th at th e ev olu tion is time-r ev er s ib le.
2. 3 M eas urement and p robabilit y
F r om th e s econ d p os tu late w e h a v e s een th at th e p os s ib le ou tcomes of a meas u r emen t ar e th e eigen v alu es of th e op er ator cor r es p on d in g to th e meas u r ed ob s er v ab le. Th e q u es tion of wh ic h on e of th e eigen v alu e w e will ob tain is s till op en . Th is q u es tion is r es olv ed b y th e th ir d p os tu late, wh ic h giv es a r ecip e on h o w to p r ed ict wh ic h ou tcome will b e ob s er v ed . Ho w ev er , th is math ematical r ecip e d o es n ot tell u s (in gen er al) with ab s olu te cer tain t y wh ic h on e of th e eigen v alu e will b e th e ou tcome. It on ly p r o v id es u s with th e p r ob ab ilit y of ob tain in g eac h eigen v alu e.
In p os tu late 3.b w e con s id er an ob s er v ab le A an d a s y s tem in th e s tate | ψ ) . Th e eigen v alu e eq u ation for th e op er ator
A (cor r es p on d in g to A ) can b e ex p r es s ed as
A | n ) = a n | n )
|( | )|
| )
wh er e a n ar e th e eigen v alu es an d n th e eigen v ector s . Th e p os tu late s tates th at th e p r ob ab ilit y of ob tain in g a n as th e ou tcome of th e meas u r emen t of A is p ( a n ) = n ψ 2 .
( | )
W e w an t to r e-ex p r es s th e p os tu late in ter ms of th e w a v efu n ction ψ ( r x ). T o d o s o, w e n eed to d efi n e th e in n er p r o d u ct in th e Hilb er t s p ace of th e w a v efu n ction s . G iv en t w o w a v e fu n ction s ψ ( r x ) an d ϕ ( r x ), th e in n er p r o d u ct ϕ ψ is giv en b y :
( ϕ | ψ ) = ∫ d 3 r x ϕ ( x ) ∗ ψ ( x )
(wh er e ∗ in d icates th e comp lex con ju gate).
W e fi r s t r ewr ite th e eigen v alu e p r ob lem for th e op er ator A in ter ms of th e eigen fu n ction s u n ( r x ) an d th e as s o ciated eigen v alu es a n :
A [ u n ( r x )] = a n u n ( r x )
Th en , w e h a v e s een th at an y fu n ction can b e ex p r es s ed in ter ms of th e eigen fu n ction s u n ( r x ). W e can as w ell ex p r es s th e w a v efu n ction in ter ms of th es e eigen fu n ction s :
Σ
ψ ( r x ) = c n u n ( r x ) , with
n
c n = ∫
d 3 r x u n ∗ ( r x ) ψ ( r x )
F in ally , accor d in g to p os tu late 3.b th e p r ob ab ilit y of ob tain in g th e ou tcome a n if th e s y s tem is in th e s tate ψ ( r x ) is giv en b y th e in n er p r o d u ct :
p ( a n ) = d 3 r x u ∗ ( r x ) ψ ( r x ) = | c n | 2
e or th ogon alit y of th e eigen fu n ction s J
∫
n
2
wh er e th e las t eq u alit y follo ws fr om th
d 3 r x u ∗ n ( r x ) u m ( r x ) = 0, for m / = n .
| |
S in ce c n 2 = p ( a n ) is a p r ob ab ilit y , th e co efficien ts c n of th e w a v efu n ction ex p an s ion ar e called pr ob abil ity am pl itu des . W e n o w con fi r m th at th e w a v efu n ction con tain all in for mation ab ou t th e s tate of th e s y s tem, s in ce giv en th e w a v e- fu n ction w e can calcu late all th e p r ob ab ilities of eac h ou tcome for eac h p os s ib le ob s er v ab le with th e follo win g p r o ce d u r e:
1) F in d th e eigen fu n ction s of th e ob s er v ab le’s op er ator . E.g., giv en an op er ator O , w e will calcu late its eigen fu n ction s
w n ( x ), s u c h th at O [ w n ( x )] = o n w n ( x ).
J
2) F in d th e p r ob ab ilit y amp litu d e of th e w a v efu n ction with r es p ect to th e eigen fu n ction of th e d es ir ed eigen v alu e ou tcome.
E.g. , if th e ou tcome is o m , s u c h th at O [ w m ( x )] = o m w m ( x ) w e will calcu late c m = d 3 r x w m ∗ ( r x ) ψ ( r x ).
3) Th e p r ob ab ilit y of ob tain in g th e giv en eigen v alu e in th e meas u r emen t is th e p r ob ab ilit y amp litu d e mo d u lu s s q u ar e. E.g. p ( o m ) = | c m | 2 .
2.3.1 W avefunc ti on c ol l aps e
Th e th ir d p os tu late s tates als o th at after th e meas u r emen t th e s y s tem is left in th e eigen s tate cor r es p on d in g to th e eigen v alu e fou n d (mor e gen er ally , if mor e th an on e eigen s tate is as s o ciated to th e s ame eigen v alu e, th e s tate is p r o jected on th e s u b s p ace of th e eigen v alu e a n , th at is , th e s u b s p ace s p an n ed b y all th e eigen s tates as s o ciated with a n ).
Th is is th e w avefu nction c ol l aps e , a con cep t th at is u s u ally q u ite p u zzlin g in q u an tu m mec h an ic s . W e can mak e th is s tatemen t at leas t a b it les s p u zzlin g b y tak in g in to accou n t th e follo win g t w o con s id er ation s .
Th e w a v efu n ction collap s e is p u zzlin g b ecau s e it p r ed icts an in s tan tan eou s ev olu tion of th e s y s tem fr om its p r e meas u r emen t s tate ψ ( x ) to its p os t-meas u r emen t s tate u n ( x ) (wh en w e meas u r e a n ). Th is t y p e of ev olu tion is v er y d iff er en t th an th e u s u al ev olu tion p r ed icted b y th e fou r th p os tu late (th at w e will s ee in a later lectu r e). Ho w ev er , th is w eir d b eh a v ior ar is es fr om con s id er in g th e meas u r emen t ap p ar atu s (an d h en ce th e meas u r emen t) as a clas s ical s y s tem, ou ts id e th e r ealm of q u an tu m mec h an ics . Alth ou gh th is v iew giv es mos t of th e time a cor r ect an s w er – an d th u s w e will u s e it in th is clas s – it is a q u ite imp r ecis e d es cr ip tion . M or e r igor ou s d es cr ip tion s of th e meas u r emen t p r o ces s , in v ok in g for ex amp le d ecoh er en ce 4 , can giv e a b etter p ictu r e of wh at actu ally h ap p en s (e.g. th e w a v e-fu n ction collap s e can tak e a fi n ite time an d b e meas u r ed ex p er imen tally in s ome cas es ).
Σ
M or e p r agmatically , th e w a v efu n ction collap s e is n eed ed in or d er to mak e ex p er imen t con s is ten t. W h at th e collap s e en tails is th at if I mak e a meas u r emen t an d I ob tain as an ou tcome th e eigen v alu e a n , I can c h ec k th at r es u lt again , b y r ep eatin g th e meas u r emen t ju s t after th e fi r s t on e (with n o time for an y c h an ge in th e s y s tem b et w een th e t w o meas u r emen t). If I cou ld n ot mak e th is s econ d c h ec k , I cou ld n ev er b e ab le to b e con fi d en t th at I got th e cor r ect an s w er th e fi r s t time (e.g. m y d etector cou ld b e wr on g) an d s o I cou ld n ev er gain an y k n o wled ge at all on m y s y s tem. O b s .: I w an t to clar ify th e mean in g of “s u b s p ace of th e eigen v alu e a n ”. If th er e is a s et of eigen s tates as s o ciated with th e eigen v alu e a n , | n j ) , th en th e s tate | ψ ) is p r o jected on to a s u p er p os ition of th es e eigen s tates | ψ ) → | n ) = j c j | n j ) .
2.3.2 P os i ti on meas urement
—
W e h a v e alr ead y calcu lated th e eigen v alu es an d eigen fu n ction s of th e p os ition op er ator . Th e eigen fu n ction s w er e u n ( x ) = δ ( x x n ) with eigen v alu es x n . W e als o calcu lated th e ex p an s ion of a fu n ction in ter ms of th e p os ition eigen fu n ction s . Rep eatin g th e calcu lation for th e w a v efu n ction w e fi n d :
c ( x n ) = ∫
dx δ ( x − x n ) ψ ( x ) = ψ ( x n ) ,
fr om wh ic h w e ob tain th at th e p r ob ab ilit y of fi n d in g a p ar ticle in th e p os ition x n is giv en b y :
p ( x n ) = | ψ ( x n ) | 2
M or e gen er ally , s in ce x is con tin u ou s , w e can d r op th e s u b s cr ip t n , as an y v alu e of x is an eigen v alu e. Th en , gen er alizin g to th e 3D cas e, w e can s a y th at th e p r ob ab ilit y of fi n d in g a p ar ticle d es cr ib ed b y th e w a v efu n ction ψ ( r x ) at th e p os ition r x is giv en b y th e mo d u lu s s q u ar e of th e w a v efu n ction its elf:
p ( r x ) = | ψ ( r x ) |
2
W e can als o s a y th at th e w a v efu n ction is th e p r ob ab ilit y amp li tu d e for th e p os ition meas u r emen t. M or e p r ecis ely , w e s h ou ld s a y th at th e p r ob ab ilit y of fi n d in g a p ar ticle b et w een x an d x + dx is p ( x ) dx = | ψ ( x ) | 2 dx wh ile | ψ ( x ) | 2 is a p r ob ab ilit y d en s it y (p er u n it len gth ). In 3D , | ψ ( r x ) | 2 is th e p r ob ab ilit y d en s it y p er u n it v olu me an d th e p r ob ab ilit y is giv en b y | ψ ( r x ) | 2 d 3 x .
4 Dec o h er en c e is th e p h en o men o n b y w h ic h a n o p en qu a n tu m s ys tem, in t er a c tin g w ith th e en vir o n men t, u n d er g o es a n ir r ev er s ib le ev o lu tio n th a t o ften lea v es it in a s ta te b es t d es c r ib ed b y th e r u les o f c la s s ic a l mec h a n ic s .
G iv en th is in ter p r etation of th e w a v efu n ction , it b ecomes n atu r al to r eq u ir e th at th e w a v efu n ction b e n or malized . W e r eq u ir e th at in tegr atin g th e p r ob ab ilit y of a p ar ticu lar p os ition o v er all p os s ib le p os ition w e ob tain 1 (i.e. cer tain t y , th e p ar ticle h as to b e s omewh er e!). Th en
∫ d 3 r x p ( r x ) = 1 → ∫ d 3 r x | ψ ( r x ) | 2 = 1
F r om b ein g a v er y ab s tr act n otion , th e w a v efu n ction h as th u s as s u med a v er y p h y s ical mean in g. W e can s till s a y th at th e w a v efu n ction d es cr ib es th e s tate of a s y s tem an d con tain s all th e in for mation ab ou t th is s y s tem. In ad d ition , an d mor e con cr etely , th e ab s olu te v alu e of th e w a v efu n ction tells u s wh er e it is mor e p r ob ab le to fi n d th e s y s tem.
2.3.3 Momentum meas urement
W e calcu lated th e eigen v alu es an d eigen fu n ction s of th e momen tu m op er ator to b e
p = i k an d u k ( x ) = Υ e ik x
J J
(Notice th at w e cou ld h a v e lab eled th e w a v e-n u m b er s as k n —an d th e momen tu m p n — to h a v e th e eigen v alu e eq u ation : p ˆ u n = p n u n , b u t w e omitted th e s u b s cr ip t n s in ce momen tu m is a con tin u ou s v ar iab le; th en w e als o s imp ly lab el th e eigen fu n ction s b y k in s tead of n ).
√ k √
| | ∞
As u s u al, w e w ou ld lik e th e eigen fu n ction s to b e n or malized . Ho w ev er n otice th at u ∗ k ( x ) u k ( x ) dx = Υ 2 dx = , s o w e can n ot fi x Υ s u c h th at th e r es u lt is n or malized as u s u al. F or con v en tion w e s et Υ = 1 : u ( x ) = 1 e ik x
2 π 2 π
(w e ar e con s id er in g th e 1D cas e). No w w e can calcu late th e p r ob ab ilit y amp litu d e for th e momen tu m meas u r emen t, b y calcu latin g th e co efficien ts of th e ex p an s ion of th e w a v efu n ction in ter ms of th e momen tu m eigen fu n ction s b as is . Her e w e r en ame th e co efficien ts c ( k ) of th e ex p an s ion ϕ ( k ). Th is is giv en b y :
c ( k ) ≡ ϕ ( k ) = u k ( x ) ψ ( x ) → ϕ ( k ) = √ 2 π
e
∫ ∗ 1 ∫
− ik x
ψ ( x )
Notice th at th is las t eq u ation is s imp ly s tatin g th at th e p r ob ab ilit y amp litu d e for th e momen tu m meas u r emen t is th e F ou r ier tr an s for m of th e w a v efu n ction , ϕ ( k ) = F [ ψ ( x )]. Th en
p ( k → k + dk ) = | ϕ ( k ) | 2 dk
is th e p r ob ab ilit y of fi n d in g th at th e p ar ticle h as a momen tu m b et w een i k an d i ( k + dk ) wh en meas u r in g th e momen tu m.
A . Flu x of p a rticles
Th e c h oice of th e co efficien t Υ imp lies th at:
∫
u ∗ k ( x ) u k ′ ( x ) dx = δ ( k − k ′ )
wh ic h is n ot th e u s u al n or malization for th e w a v efu n ction .
W h y is it n ot p os s ib le to n or malize th e momen tu m eigen s tates ?
W e s a w th at for th e w a v efu n ction , th e n or malization w as r elated to its in ter p r etation as th e p r ob ab ilit y amp litu d e for th e p os ition . If a w a v efu n ction is in a momen tu m eigen s tate ψ ( x ) = u k ( x ), th en w e can n ot r eally talk ab ou t a p ar ticle, b u t r ath er th e s y s tem is b etter d es cr ib ed b y a w a v e. In fact, th e p r ob ab ilit y of fi n d in g th e s y s tem at an y p os ition x is con s tan t an d eq u al to Υ . Th u s th e co efficien t Υ can b e b etter lin k ed to a fl u x of p ar ticles r ath er th an a p ar ticle d en s it y .
W e can s et v | ψ | 2 = Γ wh er e v = p = k k is th e v elo cit y an d Γ cor r es p on d to a fl u x of p ar ticle, as d es cr ib ed b y th e
m
.
p lan e w a v e e ik x . Th en k k | Υ | 2 = Γ fi x es th e v alu e of Υ to Υ =
m Γ
k k
m m �
2.3.4 Exp ec tati on val ues
W e h a v e ju s t s een th at th e ou tcome of a meas u r emen t is a r an d om q u an tit y (alth ou gh it is of cou r s e limited to a giv en s et of v alu es –th e eigen v alu es – fr om wh ic h w e can c h o os e fr om). In or d er to k n o w mor e ab ou t th e s tate of a s y s tem w e n eed th en to r ep eat th e meas u r emen t s ev er al times , in or d er to b u ild a s tatis tics of th e ob s er v ab le.
F or ex amp le, w e cou ld b e in ter es ted in k n o win g wh at is th e a v er age of th e meas u r emen ts of a p ar ticu lar ob s er v ab le. Th is q u an tit y is u s u ally called in Q M th e ex p ectati on v al u e of an ob s er v ab le.
Ho w d o w e u s u ally calcu late th e a v er age of a giv en q u an tit y ? Con s id er for ex amp le th e a v er age n u m b er ob tain ed b y th r o win g a d ice. In an ex p er imen t, I w ou ld h a v e to r ep eated ly th r o w th e d ice, r ecor d th e n u m b er th at comes ou t
N
i =1
n u m b er n ap p ear s an d calcu late th e a v er age n u m b er fr om th e fr eq u en cies ν n = t n / N : ( n ) = Σ 6
ν n n . In th e limit
(s a y n i ) an d th en calcu late th e s u m: ( n ) = 1 Σ N n i . Eq u iv alen tly , I cou ld cou n t th e n u m b er of times t n th at eac h
n = 1
6
→ ∞ → ∀
of N , th e fr eq u en cies ν n ap p r oac h th e p r ob ab ilities ν n p n . Th en for th e d ice w e h a v e p n = 1 / 6 ( n ) an d th e a v er age is ju s t calcu lates fr om th e s u m 1 (1 + 2 + . . . 6) = 3 . 5.
Th e p r o ced u r e for calcu latin g th e a v er age (or ex p ectation v alu e) of a q u an tit y is v er y gen er al. W e h a v e for d is cr ete an d con tin u ou s p r ob ab ilit y d is tr ib u tion fu n ction s r es p ectiv ely
( x ) = Σ p i x i ( x ) = ∫
i
dx p ( x ) x
In Q M w e ju s t n eed to r ep lace th e p r ob ab ilit y p b y its v alu e as giv en b y th e th ir d p os tu late. F or ex amp le, th e ex p ectation v alu e of th e p os ition can b e ex p r es s ed in ter ms of th e p r ob ab ilit y d en s it y fu n ction giv en b y th e mo d u lu s s q u ar e of th e w a v efu n ction :
( x ) = x |
∫
∞
ψ ( x , t )
2
− ∞
| dx
Ho w can w e in p r actice ob tain th is ex p ectation v alu e? (th at is, b y p er for min g r eal ex p er imen ts ). If w e mak e a fi r s t meas u r emen t on a s in gle p ar ticle an d th en r ep eat th e meas u r emen t o v er an d o v er again th is is n ot wh at w e meas u r e. In fact, in th at cas e w e k n o w (fr om th e p os tu lates ) th at after th e fi r s t meas u r emen t w e ex p ect alw a y s to get th e s ame r es u lt. In fact, th es e r ep eated s u cces s iv e meas u r emen t ar e on ly a w a y to c h ec k th at y es , w e got th e fi r s t an s w er cor r ect. (oth er wis e w e cou ld n ev er b e cer tain of an y th in g, s in ce w e w ou ld n ot ev en k n o w th at ou r ex p er imen tal ap p ar atu s w or k s ).
In s tead wh at w e can d o is ad op t on e of t w o s tr ategies . Eith er w e can r ep eat th e s ame ex p er im ent (n ot meas u r emen t) man y times on th e s ame s y s tem. Th is imp lies fi r s t p r ep ar in g th e s y s tem in a giv en s tate with a r ep r o d u cib le p r o ced u r e an d th en p er for min g a meas u r emen t. Th e s econ d op tion is to mak e th e ex p er imen t on a s et (an ens em bl e ) of id en tical s y s tems . In or d er to ob tain th e ex act ex p ectation v alu e w e w ou ld n eed an in fi n ite n u m b er (of r ep etition s or s y s tems ), h o w ev er a lar ge en ou gh s amp le is u s u ally p r actically en ou gh .
O b s .: Notice th at w e can r ewr ite th e ex p r es s ion ab o v e for th e ex p ect ation v alu e of th e p os ition as
∫ ∫
( x ) = ∞ ψ ( x , t ) ∗ x ψ ( x , t ) dx = ∞ ψ ( x , t ) ∗ x ˆ [ ψ ( x , t )] dx
− ∞ − ∞
Th is las t for m is a mor e gen er al on e th at is v alid for an y op er ator .
D : Exp ec t at i on val ue Th e ex p ectation v alu e of an ob s er v ab le O ˆ is
� �
O = (
ˆ
ψ | O |
ˆ
ψ ) = ψ ( x , t ) O [ ψ ( x , t )] dx
∫
∞
∗
ˆ
− ∞
∫ ∫ −
wh er e w e fi r s t u s ed th e D ir ac n otation to ex p r es s th e ex p ectati on v alu e, wh ic h is an ev en mor e gen er al ex p r es s ion . Ex amp le – Th e ex p ectation v alu e for th e momen tu m op er ator is giv en b y :
∞ ψ ( x , t ) ∗ p ˆ [ ψ ( x , t )] dx = ∞ ψ ( x , t ) ∗ i i ∂ ψ ( x , t ) dx
− ∞ − ∞ ∂ x
2. 4 Energy eigenvalue p roblem
Th e en er gy op er ator is called Hamilton ian . Th e fi r s t p os tu late s tated th at th e time d ep en d en ce of th e w a v efu n ction is d ictated b y th e S c h r ¨ od in ger eq u ation :
H
i i ∂ ψ ( r x , t ) = ψ ( r x , t )
∂ t
If w e as s u me th at ψ ( r x , t ) is th e p r o d u ct of a time-d ep en d en t p ar t T ( t ) an d a time-in d ep en d en t on e ϕ ( r x ), w e can attemp t to s olv e th e eq u ation b y s ep ar ation of v ar iab les . F r om ψ ( r x , t ) = T ( t ) ϕ ( r x ), w e can r ewr ite th e S c h r ¨ od in ger eq u ation (u s in g th e fact th at H d o es n ot c h an ge T ( t )):
∂ t ∂ t
i i ∂ T ( t ) ϕ ( r x ) = H [ T ( t ) ϕ ( r x , t )] → ϕ ( r x ) · i i ∂ T ( t ) = T ( t ) · H [ ϕ ( r x , t )]
an d w e r ear r an ge ter ms b as ed on th eir d ep en d en ce on t or r x
H
1 i i ∂ T ( t ) = 1 [ ϕ ( r x , t )] T ( t ) ∂ t ϕ ( r x )
Eac h s id e h as to b e eq u al to a con s tan t, in or d er for th e eq u alit y to h old . Th en th e time-in d ep en d en t w a v efu n ction ob ey s th e time-in d ep en d en t S c h r ¨ od in ger eq u ation :
H ϕ ( r x ) = E ϕ ( r x )
wh er e E is id en tifi ed as th e en er gy of th e s y s tem. If th e w a v efu nction is giv en b y ju s t its time-in d ep en d en t p ar t, ψ ( r x , t ) = ϕ ( r x ), th e s tate is s tationar y . Th u s , th e time-in d ep en d en t S c h r ¨ od in ger eq u ation allo ws u s to fi n d s tation ar y s tates of th e s y s tem, giv en a cer tain Hamilton ian .
Notice th at th e time-in d ep en d en t S c h r ¨ od in ger eq u ation is n oth in g els e th an th e eigen v alu e eq u ation for th e Hamil ton ian op er ator . It is th u s p ar ticu lar ly in ter es tin g to s tu d y eigen v alu es an d eigen fu n ction s of th is op er ator (wh ic h , as s aid , cor r es p on d to th e en er gies an d s tation ar y s tates of th e s y s tem) 5 .
In gen er al, th e w a v efu n ction d es cr ib in g th e s tate of a q u an tu m s y s tem is n ot
Th e en er gy of a p ar ticle h as con tr ib u tion s fr om th e k in etic en er gy as w ell as th e p oten tial en er gy :
x y z
E = 1 ( p 2 + p 2 + p 2 ) + V ( x , y , z )
2 m
In q u an tu m mec h an ics w e can fi n d th e eq u iv alen t op er ator b y s u b s titu tin g th e q u an tu m op er ator s for th e p os ition an d momen tu m in th e ab o v e ex p r es s ion :
or mor e ex p licitly :
1 2 2 2
H = 2 m ( p ˆ x + p ˆ y + p ˆ z ) + V ( x ˆ , y ˆ , z ˆ )
i 2 ∂ 2 ∂ 2 ∂ 2
an d in a comp act for m
H = − 2 m ∂ x 2 + ∂ y 2 + ∂ z 2
+ V ( x , y , z )
i 2
H = − 2 m ∇ + V ( x , y , z )
2
(Notice th at V ( x , y , z ) is ju s t a m u ltip licativ e op er ator , in th e s ame w a y as th e p os ition is ).
5 I w a n t to c la r ify th e d is tin c t io n b et w een eig en fu n c tio n s a n d w a v efu n c tio n s . I n th is c la s s w e a re in teres ted in b o th . T h e eig en fu n c tio n s a r e r ela t ed t o a g iv en o p er a t o r , a n d t h ey a r e th e s o lu tio n s to th e eig en v a lu e equ a tio n fo r th a t o p er a to r . T h ey a r e imp o r ta n t s in c e th ey fo r m a b a s is a n d t h ey a llo w u s t o c a lc u la te t h e p r o b a b ilit y o f o b ta in in g a g iv en mea s u r emen t o u tc o me. T h e w a v efu n c tio n d es c r ib es th e s ta t e o f th e qu a n tu m s ys tem. I n g en er a l, it is n o t a n eig en fu n c tio n . H o w ev er , i f w e a r e c o n s id er in g a s ta ti o na r y s ta te, th e w a v efu n c tio n th a t r ep r es en ts it m u s t b e a n eig en fu n c tio n o f th e H a milto n ia n (en er g y) o p er a to r . T h u s in th a t p a r tic u la r c a s e o n ly (w h ic h is a qu ite c o mmo n c a s e!) t h e w a v efu n c tio n is a ls o a n eig en fu n c tio n .
2.4.1 F ree pa rti c l e
In 1D , for a fr ee p ar ticle th er e is n o p oten tial en er gy , b u t on ly k in etic en er gy th at w e can r ewr ite as :
1 2 i 2 ∂ 2
H = 2 m p = − 2 m ∂ x 2
Th e eigen v alu e p r ob lem H w n ( x ) = E n w n ( x ) is th en th e d iff er en tial eq u ation
i 2 ∂ 2 w n ( x )
w e r ewr ite it as :
H w n ( x ) = E n w n ( x ) → − 2 m ∂ x 2
∂ 2 w n ( x ) 2 mE n ∂ 2 w n ( x )
= E n w n ( x )
2
wh er e w e u s ed th e id en tit y
∂ x 2
+ i 2
w n = 0 →
∂ x 2
+ k n w n = 0
i 2 k 2
n = E n
2 m
b et w een th e k in etic en er gy eigen v alu e E n an d th e w a v en u m b er k n ( an d th e momen tu m p n = i k n ).
F or a fr ee p ar ticle th er e is n o r es tr iction on th e p os s ib le en er gies , E n can b e an y p os itiv e n u m b er . Th e s olu tion to th e eigen v alu e p r ob lem is th en th e eigen fu n ction :
w n ( x ) = A s in ( k n x ) + B cos ( k n x ) = A ′ e ik n x + B ′ e − ik n x
±
W e s ee th at th er e ar e t w o in d ep en d en t fu n ction s for eac h eigen v alu e E n . Als o th er e ar e t w o d is tin ct momen tu m eigen v alu es k n for eac h en er gy eigen v alu e, wh ic h cor r es p on d to t w o d iff er en t d ir ection s of p r op agation of th e w a v e fu n ction e ± ik n x .
2. 5 Op erato rs , Commutato rs and U nc ertaint y P rinc iple
2.5.1 Commutato r
Ð —
: Commut at o r Th e Comm u tator of t w o op er ator s A , B is th e op er ator C = [ A, B ] s u c h th at C = AB B A . Ex amp le 1 – If th e op er ator s A an d B ar e s calar op er ator s (s u c h as th e p os ition op er ator s ) th en AB = B A an d th e comm u tator is alw a y s zer o.
Ex amp le 2 – If th e op er ator s A an d B ar e matr ices , th en in gen er al AB = /
B A . Con s id er for ex amp le:
A = 1 0
1 , B = 1 1 0
Th en
2 1 0
2 0 — 1
AB = 1 0 — 1 , B A = 1 0
1 ,
Th en [ A, B ] = 2 AB .
2 1 0
2 — 1 0
Ex amp le 3 – A is T u r n to you r r ight . B is T ake 3 s teps to you r l eft .
? Qu estion : d o th es e t w o op er ator s comm u te?
F ig . 1 0 : T w o r o ta tio n s A , B a lo n g th e x-a xis . Left: w e a p p ly A B (fi r s t th e 3 π / 4 r o ta tio n ) , r ig h t w e a p p ly B A . S in c e th e t w o o p er a to r s c o mm u te, th e r es u lt is th e s a me.
Ex amp le 4 – Let A an d B b e t w o r otation s . F ir s t as s u me th at A is a π / 4 r otation ar ou n d th e x d ir ection an d B a 3 π / 4 r otation in th e s ame d ir ection . No w as s u me th at th e v ector to b e r otated is in itially ar ou n d z. Th en , if w e ap p ly AB (th at mean s , fi r s t a 3 π / 4 r otation ar ou n d x an d th en a π / 4 r otation ), th e v ector en d s u p in th e n egativ e z d ir ection . Th e s ame h ap p en if w e ap p ly B A (fi r s t A an d th en B ).
F ig . 1 1 : T w o r o t a tio n s A , B a lo n g t h e x- a n d z -a xis . Left: w e a p p ly A B (fi r s t t h e π / 2 r o ta t io n a lo n g z ), r ig h t: w e a p p ly B A . S in c e t h e t w o o p er a to r s d o n o t c o mm u te, th e r es u lt is n o t t h e sa me.
No w as s u me th at A is a π / 2 r otation ar ou n d th e x d ir ection an d B ar ou n d th e z d ir ection . W h en w e ap p ly AB , th e v ector en d s u p (fr om th e z d ir ection ) alon g th e y -ax is (s in ce th e fi r s t r otation d o es n ot d o an y th in g to it), if in s tead w e ap p ly B A th e v ector is align ed alon g th e x d ir ection . In th is cas e th e t w o r otation s alon g d iff er en t ax es d o n ot comm u te.
Th es e ex amp les s h o w th at comm u tator s ar e n ot s p ecifi c of q u an tu m mec h an ics b u t can b e fou n d in ev er y d a y life. W e n o w w an t an ex amp le for Q M op er ator s .
Th e mos t famou s comm u tation r elation s h ip is b et w een th e p os i tion an d momen tu m op er ator s . Con s id er fi r s t th e 1D cas e. W e w an t to k n o w wh at is [ x ˆ , p ˆ x ] (I’ll omit th e s u b s cr ip t on th e momen tu m). W e s aid th is is an op er ator , s o in or d er to k n o w wh at it is , w e ap p ly it to a fu n ction (a w a v efu n ction ). Let’s call th is op er ator C x p , C x p = [ x ˆ , p ˆ x ].
[ x ˆ , p ˆ ] ψ ( x ) = C [ ψ ( x )] = x ˆ [ p ˆ [ ψ ( x )]] — p ˆ [ x ˆ [ ψ ( x )]] = — i i x d
— d x ψ ( x )
x p
— i i x
— dx ( x ψ ( x )) = — i i
d ψ ( x ) d
dx dx
Z x Z
— ψ ( x ) — Z x Z
= i i ψ ( x )
d ψ Z ( x Z ) d ψ Z ( x Z )
dx
dx dx
F r om [ x ˆ , p ˆ ] ψ ( x ) = i i ψ ( x ) wh ic h is v alid for all ψ ( x ) w e can wr ite
[ x ˆ , p ˆ ] = i i
{ }
Con s id er in g n o w th e 3D cas e, w e wr ite th e p os ition comp on en ts as r x , r y r z . Th en w e h a v e th e comm u tator r ela tion s h ip s :
[ r ˆ a , p ˆ b ] = i i δ a,b
th at is , v ector comp on en ts in d iff er en t d ir ection s comm u te (t h e comm u tator is zer o).
A . P rop erties of commu ta to rs
• An y op er ator comm u tes with s calar s [ A, a ] = 0
• [ A, B C ] = [ A, B ] C + B [ A, C ] an d [ AB , C ] = A [ B , C ] + [ A, C ] B
n
• An y op er ator comm u tes with its elf [ A, A ] = 0, with an y p o w er of its elf [ A, A
] = 0 an d with an y fu n ction of its elf
[ A, f ( A )] = 0 (fr om p r ev iou s p r op er t y an d with p o w er ex p an s ion of an y fu n ction ).
H
H
F r om th es e p r op er ties , w e h a v e th at th e Hamilton ian of th e fr ee p ar ticle comm u tes with th e momen tu m: [ p, ] = 0 s in ce for th e fr ee p ar ticle = p 2 / 2 m . Als o, [ x , p 2 ] = [ x , p ] p + p [ x , p ] = 2 i i p .
W e n o w p r o v e an imp or tan t th eor em th at will h a v e con s eq u en ces on h o w w e can d es cr ib e s tates of a s y s tems , b y meas u r in g d iff er en t ob s er v ab les , as w ell as h o w m u c h in for mation w e can ex tr act ab ou t th e ex p ectation v alu es of d iff er en t ob s er v ab les .
• T h e o re m: If A an d B comm u te, th en th ey h a v e a s et of n on -tr iv ial common eigen fu n ction s . Pr o of: Let ϕ a b e an eigen fu n ction of A with eigen v alu e a :
Aϕ a = aϕ a
Th en
B Aϕ a = aB ϕ a
Bu t s in ce [ A, B ] = 0 w e h a v e B A = AB . Let’s s u b s titu te in th e LHS :
A ( B ϕ a ) = a ( B ϕ a )
∝
Th is mean s th at ( B ϕ a ) is als o an eigen fu n ction of A with th e s ame eigen v alu e a . If ϕ a is th e on ly lin ear ly in d ep en d en t eigen fu n ction of A for th e eigen v alu e a , th en B ϕ a is eq u al to ϕ a at mos t u p to a m u ltip licativ e con s tan t: B ϕ a ϕ a . Th at is , w e can wr ite
B ϕ a = b a ϕ a
Bu t th is eq u ation is n oth in g els e th an an eigen v alu e eq u ation for B . Th en ϕ a is als o an eigen fu n ction of B with eigen v alu e b a . W e th u s p r o v ed th at ϕ a is a common eigen fu n ction for th e t w o op er ator s A an d B . D
Ex amp le – W e h a v e ju s t s een th at th e momen tu m op er ator comm u tes with th e Hamilton ian of a fr ee p ar ticle. Th en th e t w o op er ator s s h ou ld s h ar e common eigen fu n ction s .
Th is is in d eed th e cas e, as w e can v er ify . Con s id er th e eigen fu n ction s for th e momen tu m op er ator :
p ˆ [ ψ k ] = i k ψ k
dψ k
i i
→ — i dx = k ψ k
→ ψ k
= Ae − ik x
W h at is th e Hamilton ian ap p lied to ψ k ?
i 2 d 2 ( Ae − ik x )
i 2 k 2
− ik x
H [ ψ k ] = — 2 m dx 2
= Ae
2 m
= E k ψ k
th u s w e fou n d th at ψ k is als o a s olu tion of th e eigen v alu e eq u ation for th e Hamilton ian , wh ic h is to s a y th at it is als o an eigen fu n ction for th e Hamilton ian .
2.5.2 Commuti ng obs ervabl es
A . Degen era cy
In th e p r o of of th e th eor em ab ou t comm u tin g ob s er v ab les an d common eigen fu n ction s w e to ok a s p ecial cas e, in wh ic h w e as s u me th at th e eigen v alu e a w as non- de gener ate . Th at is , w e s tated th at ϕ a w as th e on ly lin ear ly in d ep en d en t eigen fu n ction of A for th e eigen v alu e a (fu n ction s s u c h as 4 ϕ a , α ϕ a d on ’t cou n t, s in ce th ey ar e n ot lin ear ly in d ep en d en t fr om ϕ a ).
Ð
: Degenerac y In gen er al, an eigen v alu e is d egen er ate if th er e is mor e th an on e eigen fu n ction th at h as th e s ame eigen v alu e. Th e d egen er acy of an eigen v alu e is th e n u m b er of eigen fu n ction s th at s h ar e th at eigen v alu e.
j
j
j
a a a
F or ex amp le a is n -d egen er ate if th er e ar e n eigen fu n ction { ϕ } , j = 1 , 2 , . . . , n , s u c h th at Aϕ = aϕ .
W h at h ap p en s if w e r elax th e as s u mp tion th at th e eigen v alu e a is n ot d egen er ate in th e th eor em ab o v e? Con s id er for ex amp le th at th er e ar e t w o eigen fu n ction s as s o ciated with th e s ame eigen v alu e:
Aϕ a = aϕ a an d Aϕ a = aϕ a
1 1 2 2
th en an y lin ear com b in ation ϕ a = c 1 ϕ a + c 2 ϕ a is als o an eigen fu n ction with th e s ame eigen v alu e (th er e’s an in fi n it y
1 2
of s u c h eigen fu n ction s ). F r om th e eq u alit y A ( B ϕ a ) = a ( B ϕ a ) w e can s till s tate th at ( B ϕ a ) is an eigen fu n ction of A
b u t w e d on ’t k n o w wh ic h on e. M os t gen er ally , th er e ex is t c ˜ 1 an d c ˜ 2 s u c h th at
B ϕ a = c ˜ 1 ϕ a + c ˜ 2 ϕ a
1 1 2
1
1
1
b u t in gen er al B ϕ a / ∝ ϕ a , or ϕ a is n ot an eigen fu n ction of B to o.
2 m
Ex amp le – Con s id er again th e en er gy eigen fu n ction s of th e fr ee p ar ticl e. T o eac h en er gy E = k 2 k 2 ar e as s o ciated t w o lin ear ly -in d ep en d en t eigen fu n ction s (th e eigen v alu e is d ou b ly d egen er ate). W e can c h o os e for ex amp le ϕ E = e ik x an d
±
ϕ E = e − ik x . Notice th at th es e ar e als o eigen fu n ction s of th e momen tu m op er ator (with eigen v alu es k ). If w e h ad c h os en in s tead as th e eigen fu n ction s cos ( k x ) an d s in ( k x ) th es e ar e n ot eigen fu n ction s of p ˆ .
• Th e o re m: In gen er al, it is alw a y s p os s ib le to c h o os e a s et of (lin ear ly i n d ep en d en t) eigen fu n ction s of A for th e eigen v alu e a s u c h th at th ey ar e als o eigen fu n ction s of B .
F or th e momen tu m/Hamilton ian for ex amp le w e h a v e to c h o os e th e ex p on en tial fu n ction s in s tead of th e tr igon ometr ic fu n ction s . Als o, if th e eigen v alu e of A is d egen er ate, it is p os s ib le to lab el its cor r es p on d in g eig en fu n ction s b y th e eigen v alu e of B , th u s liftin g th e d egen er acy . F or ex amp le, th er e ar e t w o eigen fu n ction s as s o ciated with th e en er gy E : ϕ E = e ± ik x . W e can d is tin gu is h b et w een th em b y lab elin g th em with th eir momen tu m eigen v alu e ± k : ϕ E , + k = e ik x an d ϕ E , − k = e − ik x .
Pro o f: A s s u me n o w w e h a v e a n e ig e n v a lu e a w ith a n n -fo ld d e g e n e ra c y s u c h th a t th e re e x is ts n in d e p e n d e n t e ig e n fu n c tio n s ϕ a ,
k = 1 , . . . , n . A n y lin e a r c o mb in a tio n o f th e s e fu n c tio n s is a ls o a n e ig e n fu n c tio n ϕ ˜ = k =1 c ˜ k ϕ k . F o r a n y o f th e s e e ig e n fu n c tio n s
(le t’s ta k e th e h t h o n e ) w e c a n w rite :
a L n a k
B [ A [ ϕ a ]] = A [ B [ ϕ a ]] = aB [ ϕ a ]
h h h
s o th a t ϕ ¯ a = B [ ϕ a ] is a n e ig e n fu n c tio n o f A w ith e ig e n v a lu e a . T h e n th is fu n c tio n c a n b e w ritte n in te rms o f th e { ϕ a } :
h h k
h
h
k
B [ ϕ a ] = ϕ ¯ a = L c ¯ h ,k ϕ a
k
T h is n o ta tio n ma k e s it c le a r th a t c ¯ h ,k is a te n s o r (a n n × n ma trix ) o p e ra tin g a tra n s fo rma tio n fro m a s e t o f e ig e n fu n c tio n s o f A
(c h o s e n a rb itra rily ) to a n o th e r s e t o f e ig e n fu n c tio n s . W e c a n w rite a n e ig e n v a lu e e q u a tio n a ls o fo r th is te n s o r,
h
c ¯ v j = b j v j → L c ¯ h ,k v j = b j v j
h
w h e re th e e ig e n v e c to rs v j a re v e c to rs o f le n g th n .
j
h
h
h
j
If w e n o w d e fi n e th e fu n c tio n s ψ a = L v j ϕ a , w e h a v e th a t ψ a a re o f c o u rs e e ig e n fu n c tio n s o f A w ith e ig e n v a lu e a . A ls o
n
j
h h
h
k
B [ ψ a ] = L v j B [ ϕ a ] = L v j L c ¯ h ,k ϕ a
h h k =1
k
h
k
k
k k
j
= L ϕ a L c ¯ h ,k v j = L ϕ a b j v j = b j L v j ϕ a = b j ψ a
k h k k
W e h a v e th u s p ro v e d th a t ψ a a re e ig e n fu n c tio n s o f B w ith e ig e n v a lu e s b j . T h e ψ a a re s imu lta n e o u s e ig e n fu n c tio n s o f b o th A a n d
j j
j
C o n s id e r th e s e t o f fu n c tio n s { ψ a } . F ro m th e p o in t o f v ie w o f A th e y a re n o t d is tin g u is h a b le , th e y a ll h a v e th e s a me e ig e n v a lu e s o th e y a re d e g e n e ra te . T a kin g in to a c c o u n t a s e c o n d o p e ra to r B , w e c a n lift th e ir d e g e n e ra c y b y la b e lin g th e m w ith th e in d e x j c o rre s p o n d in g to th e e ig e n v a lu e o f B ( b j ).
Ex a mp le – A s s u me th a t w e c h o o s e ϕ 1 = s in ( k x ) a n d ϕ 2 = c o s ( k x ) a s th e d e g e n e ra te e ig e n fu n c tio n s o f H w ith th e s a me e ig e n v a lu e
2 m
k
E = k 2 k 2 . W e n o w w a n t to fi n d w ith th is me th o d th e c o mmo n e ig e n fu n c tio n s o f p ˆ . W e fi rs t n e e d to fi n d th e ma trix c ¯ (h e re a 2 × 2
ma trix ), b y a p p ly in g p ˆ to th e e ig e n fu n c tio n s .
p ˆ ϕ 1
a n d p ˆ ϕ 2 = i i k ϕ 1 . T h e n th e ma trix c ¯ is :
= − i i d ϕ 1 = i i k c o s ( k x ) = − i i k ϕ d x 2
c ¯ = ,
0 i i k
,
− i i k 0
w ith e ig e n v a lu e s b j = ± i k , a n d e ig e n v e c to rs (n o t n o rma liz e d )
1 1
v 1 = � − i � , v 2 = � i �
W e th e n w rite th e ψ e ig e n fu n c tio n s :
ψ 1 = v 1 ϕ 1 + v 1 ϕ 2 = − i s in ( k x ) + c o s ( k x ) ∝ e − i k x , ψ 2 = v 2 ϕ 1 + v 2 ϕ 2 = i s in ( k x ) + c o s( k x ) ∝ e i k x
1 2 1 2
B . Comp lete set of commu tin g ob serva b les
{ }
| )
W e h a v e s een th at if an eigen v alu e is d egen er ate, mor e th an on e eigen fu n ction is as s o ciated with it. Th en , if w e meas u r e th e ob s er v ab le A ob tain in g a w e s till d o n ot k n o w wh at th e s tate of th e s y s tem after th e meas u r emen t is . If w e tak e an oth er ob s er v ab le B th at comm u tes with A w e can meas u r e it an d ob tain b . W e h a v e th u s acq u ir ed s ome ex tr a in for mation ab ou t th e s tate, s in ce w e k n o w th at it is n o w in a common eigen s tate of b oth A an d B with th e eigen v alu es a an d b . S till, th is cou ld b e n ot en ou gh to fu lly d efi n e th e s tate, if th er e is mor e th an on e s tate ϕ ab . W e can th en lo ok for an oth er ob s er v ab le C , th at comm u tes with b oth A an d B an d s o on , u n til w e fi n d a s et of ob s er v ab les s u c h th at u p on meas u r in g th em an d ob tain in g th e eigen v alu es a, b, c , d, . . . th e fu n ction ϕ abc d ... is u n iq u ely d efi n ed . Th en th e s et of op er ator s A, B , C , D , . . . is called a comp lete s et of comm u tin g ob s er v ab les . Th e eigen v alu es a, b, c , d, . . . th at s p ecify th e s tate ar e called go o d qu antu m nu m b er s an d th e s tate is wr itten in D ir ac n otation as a b c d . . . .
O b s . Th e s et of comm u tin g ob s er v ab le is n ot u n iq u e.
2.5.3 U nc ertai nt y p ri nc i pl e
A . Un certa in t y fo r w a ves
Th e u n cer tain t y p r in cip le, wh ic h y ou p r ob ab ly alr ead y h ear d of, is n ot fou n d ju s t in Q M . Con s id er for ex amp le th e p r op agation of a w a v e. If y ou s h ak e a r op e r h y th mically , y ou gen er ate a s tation ar y w a v e, wh ic h is n ot lo calized (wh er e is th e w a v e??) b u t it h as a w ell d efi n ed w a v elen gth (an d th u s a momen tu m).
5
10
15
20
25
x (feet)
Image by MIT OpenCourseWare.
F ig . 1 2 : A w a v e w ith a w ell d efi n ed w a v elen g t h b u t n o w ell-d efi n ed p o s it io n .
k
If in s tead y ou giv e a s u d d en jer k , y ou cr eate a w ell lo calized w a v ep ac k et. No w h o w ev er th e w a v elen gth is n ot w ell d efi n ed (s in ce w e h a v e a s u p er p os ition of w a v es with man y w a v elen gth s ). Th e p os ition an d w a v elen gth can n ot th u s b e w ell d efi n ed at th e s ame time. In Q M w e ex p r es s th is fact with an in eq u alit y in v olv in g p os ition an d momen tu m
λ
x
p
p = 2 π k . Th en w e h a v e σ σ ≥ . W e ar e n o w goin g to ex p r es s th es e id eas in a mor e r igor ou s w a y .
2
5
10
15
20
25
x (feet)
Image by MIT OpenCourseWare.
F ig . 1 3 : A w a v e p a c k et w ith a w ell d efi n ed p o s itio n b u t n o w ell-d efi n ed w a v elen g th . (F r o m Gr iffith )
B . R ep ea ted mea su remen ts
Recall th at th e th ir d p os tu late s tates th at after a meas u r emen t th e w a v efu n ction c ol l aps es to th e eigen fu n ction of th e eigen v alu e ob s er v ed .
Σ
Let u s as s u me th at I mak e t w o meas u r emen ts of th e s ame op er ator A on e after th e oth er (n o ev olu tion , or time to mo d ify th e s y s tem in b et w een meas u r emen ts ). In th e fi r s t meas u r emen t I ob tain th e ou tcome a k (an eigen v alu e of A ). Th en for Q M to b e con s is ten t, it m u s t h old th at th e s econ d meas u r emen t als o giv es me th e s ame an s w er a k . Ho w
| | / /
| |
is th is p os s ib le? W e k n o w th at if th e s y s tem is in th e s tate ψ = k c k ϕ k , with ϕ k th e eigen fu n ction cor r es p on d in g to th e eigen v alu e a k (as s u me n o d egen er acy for s imp licit y ), th e p r ob ab ilit y of ob tain in g a k is c k 2 . If I w an t to imp os e th at c k 2 = 1, I m u s t s et th e w a v efu n ction after th e meas u r emen t to b e ψ = ϕ k (as all th e oth er c h , h = = k ar e zer o).
Th is is th e s o-called c ol l aps e of the w avefu nction . It is n ot a m y s ter iou s accid en t, b u t it is a p r es cr ip tion th at en s u r es th at Q M (an d ex p er imen tal ou tcomes ) ar e con s is ten t (th u s it’s in clu d ed in on e of th e p os tu lates ).
No w con s id er th e cas e in wh ic h w e mak e t w o s u cces s iv e meas u r emen ts of t w o d iff er en t op er ator s , A an d B . F ir s t w e meas u r e A an d ob tain a k . W e n o w k n o w th at th e s tate of th e s y s tem after th e meas u r emen t m u s t b e ϕ k . W e n o w h a v e t w o p os s ib ilities .
If [ A, B ] = 0 (th e t w o op er ator comm u te, an d again for s imp licit y w e as s u me n o d egen er acy ) th en ϕ k is als o an eigen fu n ction of B . Th en , wh en w e meas u r e B w e ob tain th e ou tcome b k with cer tain t y . Th er e is n o u n cer tain t y in th e meas u r emen t. If I meas u r e A again , I w ou ld s till ob tain a k . If I in v er ted th e or d er of th e meas u r emen ts , I w ou ld h a v e ob tain ed th e s ame k in d of r es u lts (th e fi r s t meas u r emen t ou tcome is alw a y s u n k n o wn , u n les s th e s y s tem is alr ead y in an eigen s tate of th e op er ator s ).
h
| |
eigen fu n ction s of B , ϕ k =
h
c k ψ h (wh er e ψ h ar e eigen fu n ction s of B with eigen v alu e b h ). A meas u r emen t of B d o es
Th is is n ot s o s u r p r is in g if w e con s id er th e clas s ical p oin t of v iew, wh er e meas u r emen ts ar e n ot p r ob ab ilis tic in n atu r e. Th e s econ d s cen ar io is if [ Σ A, B ] / = / = 0. Th en , ϕ k is n ot an eigen fu n ction of B b u t in s tead can b e wr itten in ter ms of
h
n ot h a v e a cer tain ou tcome. W e w ou ld ob tain b h with p r ob ab ilit y c k 2 .
Th er e is th en an in tr in s ic u n cer tain t y in th e s u cces s iv e meas u r emen t of t w o n on -comm u tin g ob s er v ab les . Als o, th e r es u lts of s u cces s iv e meas u r emen ts of A, B an d A again , ar e d iff er en t if I c h an ge th e or d er B , A an d B .
It mean s th at if I tr y to k n o w with cer tain t y th e ou tcome of th e fi r s t ob s er v ab le (e.g. b y p r ep ar in g it in an eigen fu n ction ) I h a v e an u n cer tain t y in th e oth er ob s er v ab le. W e s a w th at th is u n cer tain t y is lin k ed to th e comm u tator of th e t w o ob s er v ab les . Th is s tatemen t can b e mad e mor e p r ecis e.
� — ( ) � �
• T h e o re m: D efi n e C = [ A, B ] an d ΔA an d ΔB th e u n cer tain t y in th e meas u r emen t ou tcomes of A an d B : ΔA 2 =
� � Σ
A 2 A 2 , wh er e O ˆ is th e ex p ectation v alu e of th e op er ator O ˆ (th at is , th e a v er age o v er th e p os s ib le ou tcomes ,
ΔAΔB ≥ 2 | ( C ) |
1
for a giv en s tate: O ˆ = ( ψ | O ˆ | ψ ) = k O k | c k | 2 ). Th en :
Th is is Hei s en b erg Un certai n t y Pri n ci p l e .
Ex amp le – Th e mos t imp or tan t ex amp le is th e u n cer tain t y r elation b et w e en p os ition and momen tu m. W e k n o w th at th es e t w o op er ator s d o n ot comm u te an d th eir comm u tator is [ x ˆ , p ˆ ] = i i . Th en
i
Δ x ˆ Δ p ˆ ≥ 2
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22.02 Introduction to Applied Nuclear Physics
Spring 2012
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