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Chapter 5: Q18CQ (page 187)

Quantum-mechanical stationary states are of the general form 唯(x,t)=蠄(x)e-i蝇t. For the basic plane wave (Chapter 4), this is 唯(x,t)=Aeikxe-i蝇t=Aei(kx-蝇t), and for a particle in a box it is Asinkxe-i蝇t. Although both are sinusoidal, we claim that the plane wave alone is the prototype function whose momentum is pure-a well-defined value in one direction. Reinforcing the claim is the fact that the plane wave alone lacks features that we expect to see only when, effectively, waves are moving in both directions. What features are these, and, considering the probability densities, are they indeed present for a particle in a box and absent for a plane wave?

Short Answer

Expert verified

Feature for a particle in a box: the wave function is equal to zero at any point where kx=n蟺.

Feature for a plane wave: the function could never be equal to zero, hence has pure momentum and well-defined value in one direction.

Step by step solution

01

Given data

For a basic plane wave, the wave function is given by 唯(x,t)=Aei(kx-蝇t) and for a particle in a box the wave function is given by 唯(x,t)=Asin(kx)e-i蝇t.

02

Concept of quantum mechanics

In quantum mechanics, a particle does not have a fixed boundary or a definite momentum. Here, particles are considered to be a superposition of a large number of waves forming a wave packet.

03

Time-dependent wave

If a wave is moving in both directions, then it will form a standing wave, and hence nodes will be formed. For a particle in a box, the solution has the term sinkx, which is definitely zero when kx=n蟺but for a plane wave, the function expikx-蝇tis not zero, though its complex square is a constant.

Hence, we see that a wave function for a plane wave cannot be zero.

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Most popular questions from this chapter

Using equation (23), find the energy of a particle confined to a finite well whose walls are half the height of the ground-state infinite well energy, . (A calculator or computer able to solve equations numerically may be used, but this happens to be a case where an exact answer can be deduced without too much trouble.)

Because protons and neutrons are similar in mass, size, and certain other characteristics, a collective term, nucleons, has been coined that encompasses both of these constituents of the atomic nucleus. In many nuclei, nucleons are confined (by the strong force, discussed in Chapter11) to dimensions of rough15femtometers. Photons emitted by nuclei as the nucleons drop to lower energy levels are known as gamma particles. Their energies are typically in the MeVrange. Why does this make sense?

Refer to a particle of massdescribed by the wave function

蠄(x)={2a3xe-axX>00X<0

Verify that the normalization constant2a3 is correct.

Consider the delta well potential energy:

U(x)={0x鈮�0-鈭�x=0

Although not completely realistic, this potential energy is often a convenient approximation to a verystrong, verynarrow attractive potential energy well. It has only one allowed bound-state wave function, and because the top of the well is defined as U = 0, the corresponding bound-state energy is negative. Call its value -E0.

(a) Applying the usual arguments and required continuity conditions (need it be smooth?), show that the wave function is given by

蠄(x)=(2mE0h2)1/4e-(2mE0/鈩�)|x|

(b) Sketch 蠄(x)and U(x) on the same diagram. Does this wave function exhibit the expected behavior in the classically forbidden region?

Consider the wave function that is a combination of two different infinite well stationary states the nth and the mth

蠄x,t=12蠄nxe-iEn/鈩�t+12蠄me-iEm/鈩�t

  1. Show that the 蠄x,tis properly normalized.
  2. Show that the expectation value of the energy is the average of the two energies:E炉=12En+Em
  3. Show that the expectation value of the square of the energy is given by .
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