22.101 Applied Nuclear Physics (F all 2006)

Lecture 3 (9/13/06)

Bound States in One D i mensional Systems – Particle in a Square Well

References --

R. L. Liboff, Introductory Quantum Mechanics (Holden Day, New York, 1980).

We will solv e the Schröd inger wave equation by considering the sim plest problem in quantum m e chanics, a p a rticle in a po tential well. The student will see from this calculation how the problem is treated by dividing the system into two regions, the inter ior where the p a rticle feels the p o tential, and the exterio r where the p a rticle is a free particle (zero potential). The solutio ns to th e wave equation have to be different in these two regions to reflect the binding of the particle - the wave function is oscillatory in th e interior region and exponentially decaying (non-oscillatory) in the exterior region.

Matching these two solutions at the boundary where the potential goes f r om a finite value (interio r) to zero (exterio r ) gives a condition on the wavenum ber (or wavelength), which turns out to be the condition of quantization . The m eaning of quantization is that solution s exist only if th e wavenum bers take on certain discrete values, which then translate into discrete en ergy lev els f o r the pa rticle. For a given potential well of certain depth and width, only a discrete set of wave functions can exist in the p o tential well.

These wave functions are the eigenfunctions of the Ham ilton ian (en ergy) operato r, with corresponding energy levels as the eigenvalues. Finding the wa vefunctions and the spectrum of eigenvalues is what we m ean by solving the Schrödinger wave equation for the particle in a potential well. Changing the sh ape of the potential m eans a different set of eigenfunctions and the eigenvalues. The procedure to find them , howe ver, is the sam e.

For a one-dim ensional system the tim e-independent wave equation is

  ħ 2 d 2  ( x )  V ( x ) 

2 m dx 2

( x ) E 

( x ) (3.1)

We will use this equ atio n to investig ate th e boun d-states of a particle in a square well potential of depth V o and width L. The physical m eaning of (3.1) is essentially the

statem ent of energy conservation, the total energy E, a negative and constant quantity in the present problem of bound-state calculatio ns, is the sum of kinetic and potential energies. Since (3.1) holds at every point in space, the fact that th e potential energy V(x) varies in space m eans the kinetic energy of the particle also w ill v a ry in sp ace. For a square well potential, V(x) has the form

V ( x )  V o  L /2   x   L /2

= 0 elsewhere (3.2) as shown in Fig. 1. Taking advantage of the piecewise constant behavior of the potential,

Fig. 1. The square well potential centered at the origin with depth V o and width L.

we divide th e configuration space (ou r entire system ) into an interior regio n, where the potential is constant and nega tive, and an exterior region where the potential is zero. F o r the interior region the wave equation can be put into the standard form of a second-order differential equation with constan t coefficient,

d 2  ( x ) 

dx 2

k 2  ( x ) 0 x   L /2 (3.3)

In (3.3) we purposely introduced the wavenum ber k, with k 2 2 m ( E  V o )/ ħ 2 always positiv e, so that k is real. (As an asid e note th at we will be d o ing this co nsisten tly in

writing out the wave equation to be solved. In other words, the wavenumber we introduce is always real, whereas the sign of the second term i n the wave equation can be plus, as seen in (3.3), o r m i nus, as in (3.4).) Fo r k 2 to be po sitiv e we un derstand that the solution s where –E > V o will be ex cluded from our considerations.

For the ex terior reg ion, the wave equ ation sim ilarly can be pu t into the form

d 2  ( x )   2 

dx 2

( x ) 0 x

  L /2 (3.4)

where  2  2 mE / ħ 2 . To obtain the solutions of physical interest to (3.3 ) an d (3.4), we keep in m i nd that the so lutions should have certain symm etry properties, in this case they should have definite parity, or inversion symmetry (see below). This m eans that when

x  -x,  ( x ) m u st be either invariant or it must change sign. The reason for this

requirem ent is that the H a m iltonian H is symmetric under inversion (the potential is symmetric given our choice of the coordinate sy stem (see Fig. 1)). Thus we take for our solution s

 ( x ) A sin kx x   L /2

= Be   x

x > L / 2 ( 3 . 5 )

= Ce  x x < - L / 2

We have used the condition of definite parity in choosing the interior solution as a sine function, an odd function . The choice of a solution with odd parity is arbitrary b ecause an even-parity solution, coskx, would be equally acceptable. W h at about a linear com bination of the two solutions, such as the sum , Asinkx + Bcoskx? This particular choice would not be acceptable because the sum of an odd and even parity solutions violates the requirem ent that all so lutions m u st have definite parity.

For the exterior region we have applied condition (i) in Lec2 to discard the exponentially growing solution. This is justified on physical grounds; for a bound state

the particle should be m o stly localized inside the potential well, which is another way of saying th at away from the well th e wave function should be d ecaying rath er than growing.

In the so lutions we have chosen there are three constants of integration, A, B, and

C. These are to be determ ined by applyi ng boun dary conditions at th e in terface between the interior and exterior re gions, plus a norm alization condition (2.23). Notice there is another constant in the problem whic h has not been specified, the energy eigenvalue E. All we have said thus far is that E should be negative. W e have already u tilized the boundary co ndition at in finity and th e inversion symm etry condition of definite parity. The conditions which we have not yet applied are the continuity conditions (ii) and (iii) in Lec2. At the in terface, x o  L /2 , the boundary conditions are

 int ( x o )  ext ( x o ) ( 3 . 6 )

x o x o

(3.7)

with subscripts int and ext denoting the inte rior and exterior solutions respectiv ely.

The four conditions at the interface do not allow us to determ ine the four constants because our system of equa tions is homogeneous. As in situations of this kind, the proportionality constant is fixed by the norm alization co ndition (2.2 3). W e therefore obtain C = -B, B A sin( kL / 2 ) exp(  L / 2 ) , and

cot( kL / 2 )   / k ( 3 . 8 )

with the con stant A determ ined by (2.23). Eq.(3.8 )is th e m o st im portant result of this calcu lation; it is som etim es called a dispersion relation. It is a relation w h ich determ ines the allowed values of E, a quantity that appears in both k and . These are then the discrete (qu antized) energy levels which the p a rticle can hav e in the p a rticular po tential well given, a square well of width L and depth V o . Eq.(3.8) is the consequence of

choosing odd-parity solutions for the interior wave. For the even-parity solutions,

 int ( x ) A ' cos kx , the corresponding dispersion relation is

tan( kL / 2 )  / k ( 3 . 9 )

Since both solutions are equally acceptable, on e has two distinct sets of energy lev els, given im plicitly by (3.8) and (3.9). T o see what are these lev els m o re explicitly, further analysis is n ecessary.

W e consider a graphical analysis of (3.8) and (3.9 ). W e first p u t the two equations into dim ensionless form ,

 cot     (odd-parity) (3.10)

 tan   (even-p arity ) (3.11) where  kL /2 ,   L / 2 . Then we notice that

 2   2 2 mL 2 V / 4 ħ 2   ( 3 . 1 2 )

is a constant for fixed values of V o and L. In Fig. 4 we plot the left- and right-hand sides of (3.10) and (3.11), and obtain from their intersections the allowed energ y levels. The graphical method thus reveals the following features. There exists a m i nimum value of

Fig. 4. Graphical solutions of (3.10) and (3.11) showing that there could be no odd- parity solu tions if  is not large enough (the potential is not deep enough or not wide enough), while there is at least one even-parity solution no m a tter what values are the well depth and width.

 below which no odd-parity solutions are allowed. On the other hand, there is always at least one ev en-parity so lution. The firs t even-p arity energy level occu rs at    /2 , whereas the first odd-parity level occurs at  /2     . Thus, the even- and odd-

parity levels alternate in m a gnitudes, with the lowest level being even in p a rity. W e should also note that the solutions depend on the potential function parameters only through the variable  , or the com bination of V o L 2 , so that the effect of any change in well depth can be com pe nsated by a change in the square of the well width.

At this po int it can b e no ted that we anticipate that for a particle in a po ten tial well in three dim ensions (n ex t chapter), th e cosine solution to the wave function has to be discarded because of the condition of regularity (wave functi on m u st be finite) at the origin. Th is m eans that there will be a m i ni m u m value of  or V o L 2 below which no bound states can exist. This is a feature of problem s in three dim ensions which does not apply to problem s in one dim ension.

We now summarize our results for the allo wed energy lev els of a particle in a square well potential and the corresponding wave functions.

 int ( x ) A sin kx or A ' cos kx x   L / 2 (3.13)

 ( x ) Be   x x > L/2

= Ce  x x < -L/2 (3.14)

where the en ergy lev els are

ħ 2 k 2 ħ 2  2

E  V o  2 m  2 m

( 3 . 1 5 )

The constan ts B and C are determ ined from the continuity co nditions at the interface, while A and A’ are to be fixed by the norm alization condition. The discrete values of the bound-state energies, k or , are ob tain ed from (3.8) and (3.9). In Fig. 5 we show a

Figure by MIT OCW.

Fig. 5. Ground-state and first two excited-st ate solutions [from Cohen, p. 16]. Approxim ate solutions given by the condition of vanishing w a vefunction at the potential boundary are indicated by the dashed lines.

sketch of the three lowest-level solutions, the ground state with even parity, the first excited state with odd parity, and the second excited state with even parity. Notice that the num ber of excited states that one can have depends on the value of Vo because ou r solution is v alid on ly for negativ e E. This m eans that for a potential of a given depth, the particle can be bound only in a finite num ber of states.

To obtain more explicit results it is w o rthwhile to consider an approxim ation to the boundary condition at the interface. Instead of the contin uity of  and its derivative at the interface, one m i ght assum e that the p e netration of the wave function into th e

extern al region can be neglected , and therefore require that  vanishes at x  L /2 .

Applying this condition to (3.13) gives kL n  , where n is any integer, or equivalently,

E n  V o

n 2  2 ħ 2

2 mL 2 , n = 1, 2, … (3.16)

This shows explicitly ho w the energ y eigenvalue E n varies with the level index n, which is the quantum num ber for the one-dim ensional problem under consideration. The corresponding wave functions under this approxim ation are

 n ( x ) A n cos( n  x / L ) , n = 1, 3, …

= A ' sin( n  x / L ) n = 2, 4, … (3.17)

The first so lutions in this approxim ate calcula tion are also shown in Fig. 5. W e see that requiring th e wave function to van ish at th e interface has the eff ect of confining the particle in a potential well of width L with infinitely steep walls (the infinite well potential or lim it of V o  ). It is therefore to be expected that the problem becom es independent of V o and there is no lim it on th e nu m b er of excited states. C l early, the approxim ate solutions becom e the more useful th e greater is the well dep th, and the error is always an overestim ate of the energy leve ls as a result of squeezing of the wave function (ph ysically, th is m a kes the wave have a shorter period or a larger wavenum ber).