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Chapter 5: Q92CE (page 193)

a) Taking the particle鈥檚 total energy to be 0, find the potential energy.

(b) On the same axes, sketch the wave function and the potential energy.

(c) To what region would the particle be restricted classically?

Short Answer

Expert verified

The potential energy is U=h2m3x2-a2m(x2+a2)2

Step by step solution

01

Potential energy

(a) If the total energy is 0, the potential energy will be the negative kinetic energy; the kinetic energyhas an operator.

E^=-h22md2dx2

The kinetic energy is

E^(2螤a3/21x2+a2)=-h22md2dx2(2螤a3/21x2+a2)

=h22m2螤a3/2a2-3x2(x2+a2)3

=h22m2螤a3/21x2+a22a2-3x2(x2+a2)2

The double derivative is calculated separately below:

d2dx2-1x2+a2=ddx2x(x2+a2)2

(x2+a2)2-x2(x2+a2)2x(x2+a2)2=2a2-3x2(x2+a2)3

Therefore, the potential energy isU=h2m3x2-a2m(x2+a2)2.

02

Sketch of potential energy and wave function

(b) The sketch is as follows. The blue line represents the potential energy, and the red line represents the wave function.

The unit length is

03

Find the region

The classical region is where the potential energy is smaller than 0, which is about (-0.6a,0.6a).

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Most popular questions from this chapter

A student of classical physics says, "A charged particle. like an electron orbiting in a simple atom. shouldn't have only certain stable energies: in fact, it should lose energy by electromagnetic radiation until the atom collapses." Answer these two complaints qualitatively. appealing to as few fundamental claims of quantum mechanics as possible.

The potential energy shared by two atoms in a diatomic molecule, depicted in Figure 17, is often approximated by the fairly simple function U(x)=(ax12)-(bx6)where constants a and b depend on the atoms involved. In Section 7, it is said that near its minimum value, it can be approximated by an even simpler function鈥攊t should 鈥渓ook like鈥� a parabola. (a) In terms ofa and b, find the minimum potential energy U (x0) and the separation x0 at which it occurs. (b) The parabolic approximation UP(x)=U(xo)+12魏(x-xo)2has the same minimum value at x0 and the same first derivative there (i.e., 0). Its second derivative is k , the spring constant of this Hooke鈥檚 law potential energy. In terms of a and b, what is the spring constant of U (x)?

Whereas an infinite well has an infinite number of bound states, a finite well does not. By relating the well heightU0 to the kinetic energy and the kinetic energy (through 位) to n and L. Show that the number of bound states is given roughly by8ml2U0/h2 (Assume that the number is large.)

Equation (5 - 16) gives infinite well energies. Because equation (5 - 22) cannot be solved in closed form, there is no similar compact formula for finite well energies. Still many conclusions can be drawn without one. Argue on largely qualitative grounds that if the walls of a finite well are moved close together but not changed in height, then the well must progressively hold fewer bound states. (Make a clear connection between the width of the well and the height of the walls.)

Show that the uncertainty in the momentum of a ground state harmonic oscillator is 鈩�2mk1/4.
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