Particle in a rectangular well

Consider a particle in a one-dimensional box with walls of finite height (Fig. 2.5a). The potential-energy function is F = Fq for x 0, F = 0 for 0 x /, and F = Vo for X /. There are two cases to examine, depending on whether the particle s energy E is less than or greater than Vq. 

FIGURE 2.5 (a) Potential energy for a particle in a one-dimensional rectangular well, (b) The ground-state wave function for this potential, (c) The first excited-state wave function. 

As in Section 2.3, we must prevent lAi from becoming infinite as x — - . Since we are assuming E Vq, the quantity ( Vq is a real, positive number, and to keep lAi finite as x — - co, we must have D = 0. Similarly, to keep lAm finite as x we must 

To complete the problem, we must apply the boundary conditions. As with the particle in a box with infinite walls, we require the wave function to be continuous at x = 0 and at X = / so. Ai(O) = Aii(O) and. An(0 = Ani(0- The wave function has four arbitrary constants, so more than these two boundary conditions are needed. As well as requiring ij/ to be continuous, we shall require that its derivative dip/dxbe continuous everywhere. To justify this requirement, we note that if difi/dx changed discontinuously at a point, then its derivative (its instantaneous rate of change) d p/dx would become infinite at that point. However, for 

Only the particular values of F that satisfy (2.33) give a wave function that is continuous and has a continuous derivative, so the energy levels are quantized for F Vo. To find the allowed energy levels, we can plot the left side of (2.35) versus e for 0 e 1 and find the points where the curve crosses the horizontal axis (see also Prob. 4.31c). A detailed study (Merzbacher, Section 6.8) shows that the number of allowed energy levels with F Vq is N, where N satisfies 

The free-particle problem is an unreal situation because we could not actually have a particle that had no interaction with any other particle in the universe. 

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A state in which > 0 as jc —> oo and as x —oo is called a bound state. For a bound state, significant probability for finding the particle exists in only a finite region of space. For an unbound state, ip does not go to zero as x —> oo and is not normalizable. For the particle in a rectangular well, states with < Vq are bound and states with E> Vq are unbound. For the particle in a box with infinitely high walls, all states are bound. For the free particle, all states are unbound. 

For the particle in a rectangular well (Section 2.4), Fig. 2.5 and the equations for ipi and 4fui show that for the bound states there is a nonzero probability of finding the particle in regions I and III, where its total energy E is less than its potential energy V = Vq. Classically, this behavior is not allowed. The classical equations E = T + V and T > 0, where T is the kinetic energy, mean that E cannot be less than V in classical mechanics. 

For the particle in a rectangular well (Section 2.4), show that in the limit Vq ... 

For a particle in a rectangular well of depth Vq and width /, state whether the number of bound-state energy levels increases, decreases, or remains the same (a) as Vq increases at fixed / (b) as I increases at fixed Vq. 

The particle in a rectangular one-dimensional well with walls of finite height has a finite number of bound states [Eq. (2.36)]. The bound-state wave functions are oscillatory inside the well and die off exponentially to zero outside the well. The energies of the unbound states are not quantized. 

Geometrical sizes of H2 molecule and van der Waals gap width of InSe crystal have such values that cause quantum-size effects. Using the well-known expression [12] for localization of a particle in the quantum well with infinite rectangular walls... 

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