Problems in One-Dimension: General Considerations, Infinite Well Potential,
6.1 General Considerations
Chapter 6
Problems in One-Dimension: General
Fig. 6.1 Potential barriers
Motion in classical mechanics can be bounded or unbounded. Bounded, one- dimensional motion in classical mechanics is restricted to a finite region of space, while unbound motion can extend to eitherxD 1and/orxD 1. Bound states in quantum mechanics correspond to states that are, for the most part, localized to finite regions of space. In other words, a necessary condition for a bound state in quantum mechanics is that the eigenfunction associated with a bound state goes to zero asjxj ! 1(in three dimensions the eigenfunction must go to zero asr! 1).
Unbound states in quantum mechanics correspond to states whose eigenfunctions do not vanish for eitherx D 1 and/orx D 1. The eigenfunctions of the free particle do not correspond to a bound state. I now examine the classical motion and quantum-mechanical properties associated with several generic classes of one- dimensional potentials.
6.1.1 Potentials in which V.x/ > 0 and V .˙1/ 0
Potentials falling into this class are shown in Fig.6.1. In both cases shown, there are continuous eigenenergies E > 0 and a two-fold degeneracy for each energy, since particles (waves) can be incident from the left or right. Were I to solve the time-independent Schrödinger equation for potentials of this type, I would find that the eigenenergies correspond to all positive energies. Classically, in case (a), a particle having energyE1 incident from the left would be reflected by the potential. Quantum-mechanically there is some probability that the particletunnels to the other side and is transmitted through the barrier. Tunneling is a wave-like phenomenon. A classical particle having energyE2incident from the left is always
6.1 General Considerations 117 transmitted with no reflection, although its kinetic energy changes as it moves by the potential barrier. Quantum-mechanically there is some probability that the particle is reflected. For a potential that varies slowly over a de Broglie wavelength of the particle, this reflection is very small, but for a rectangular barrier it can be significant. As you shall see, the rectangular barrier case is analogous to one in which light is reflected by a dielectric slab. In case (b), classically, a particle having energyE1can be bound if it is located in the potential well. Quantum-mechanically there are no bound states, a particle prepared inside the well eventually tunnels out. A classical particle having energy E1 incident from the left would always be reflected. Quantum-mechanically there is some probability that the particle is transmitted as a result of tunneling. A classical particle having energyE2incident from the left would always be transmitted with no reflection. Quantum-mechanically there is some probability that the particle is reflected.
6.1.2 Potentials in which V.x/ > 0 and V .1/ 0 while V .1/ 1
For the potential of (Fig.6.2) (a), a classical particle is reflected by the potential, as is a quantum wave packet incident from the left. Since a wave packet cannot be incident from the left, there is no degeneracy, although the eigenenergies are continuous. In case (b) there are no bound states quantum-mechanically, although there could be bound states classically for energyE1, if the particle is located in the potential well.
Fig. 6.2 Reflecting potentials
Fig. 6.3 Step potential
6.1.3 Potentials in which V.x/ > 0 and V .1/ 0 while V .1/ D W > 0
In the case of a step potential (Fig.6.3), a wave packet cannot be incident from the right ifE < W. Therefore, forE < W, the eigenenergies are continuous, but there is no degeneracy. A wave packet incident from the left is totally reflected.
On the other hand, forE > W;there are continuous eigenenergies and a two-fold degeneracy since wave packets can be incident from the left or right. Classically, a particle incident from the left is reflected ifE < W and transmitted with reduced kinetic energy (with no reflection) ifE > W. Quantum mechanically there can be some reflection for a wave packet incident from the left having energyE>W.
6.1.4 Potentials in which V.x/ > 0 and V .˙1/ 1
For the potentials of (Fig.6.4), a wave packet can be incident neither from the right nor the left. The wave function must vanish asjxj ! 1. In order to fit the waves in the potential and satisfy this boundary condition, the energy must take on discrete values. There is no degeneracy in this case, but there is an infinite number of discrete energies possible. Classically, any energy greater than the minimum value of the potential energy is allowed. A particle could be bound in one of the sub-wells in case (b) for energyE1. Quantum-mechanically, there are no bound states that are localizedentirelyin onlyonesub-well.
6.1.5 Potentials in which V.x/ < 0 and V .˙1/ 0
In this case (Fig.6.5) of a potential well, forE> 0, a wave packet can be incident from the left or right. Therefore, forE > 0, the eigenenergies are continuous, and there is a two-fold degeneracy. A wave packet incident from the left will be partially reflected and partially transmitted. There can also beresonancephenomena, as with
6.1 General Considerations 119
Fig. 6.4 Bound state potentials
Fig. 6.5 Potential well
reflection of light from thin dielectric films. Classically there is no reflection for E> 0. ForE< 0;quantum-mechanically there is no degeneracy and there is a finite number of bound states. It can be proven that there is always at least one bound state, regardless of the depth of the potential.1This might seem a little surprising, but, for low potential depths, the wave function extends significantly into theclassically
1Somewhat more precise requirements that guarantee the existence of a bound state areV.1/D V.1/DV0and
Z 1 1
ŒV.x/V0dx< 0:
You are asked to prove this using the variational method in Problem 15.8.
forbiddenregions (regions where the kinetic energy would be negative) giving rise to a largexand a correspondingly smallpthat is sufficiently small to prevent the particle from being freed from the well. Classically a particle having energyE< 0 is bound in the well.
Other types of potentials are also possible, but you should get the idea by now.
Inbound stateone-dimensional problems,there is never energy degeneracy—the actual number of bound states (if any) depends on the details of the potential. For one-dimensional problems giving rise to continuous eigenenergies, there is two-fold degeneracy if the energy is greater than the potential asjxj ! 1.
Values ofxfor whichE>V.x/correspond to theclassically allowed regionfor the particle and to one for which Schrödinger’s equation is
d2
dx2 D k2.x/ ; k.x/D
r2mŒEV.x/
„2 > 0: (6.1) IfV is constant, the solutions are sines or cosines of kx. In general the solution is oscillatory in such classically allowed regions. On the other hand, values of x for whichE < V.x/corresponds to aclassically forbidden regionfor the particle, since its kinetic energy would have to benegative. WhenE<V.x/, Schrödinger’s equation is
d2
dx2 D2.x/ ; .x/D
r2mŒV.x/E
„2 > 0: (6.2) For constantV, the solutions are real exponentials of˙x. In general the solution is a smooth decaying function the deeper you penetrate into classically forbidden regions.