Chapter 5: Q44E (page 189)

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.)

Short Answer

Expert verified

The energy of a particle confined to a finite well whose walls are half the height of the ground-state infinite well energy is $\frac{π^{2} ℏ^{2}}{8 m L^{2}}$ .

Step by step solution

Given data

There is a finite well whose walls are half the height of the ground-state infinite well energy, E₁.

Ground state energy of infinite potential well and height of the finite potential well

The ground state energy of a particle of mass in an infinite well of width L is

$E = \frac{π^{2} ℏ^{2}}{2 m L^{2}}$ .....(I)

Here $ℏ$ is the reduced Planck's constant.

The height of the finite potential well is

$U_{0} = \{\begin{cases} \frac{ℏ^{2} k^{2}}{2 m} s e c^{2} \frac{k L}{2} & (n - 1) π < k L < n π n = 1, 3, 5 . . . \\ \frac{ℏ^{2} k^{2}}{2 m} c s c^{2} \frac{k L}{2} & (n - 1) π < k L < n π n = 2, 4, 6 . . . \end{cases}$ .....(II)

Determining the energy of the finite well

The height of the well is half the ground state energy of an infinite potential well. Thus for n = 1

$U_{0} = \frac{E_{1}}{2}$

Substitute from equations (I) and (II) to get

$\begin{array}{rcl} \frac{ℏ^{2} k^{2}}{2 m} \sec^{2} \frac{k L}{2} & = & \frac{π^{2} ℏ^{2}}{4 m L^{2}} \\ k^{2} \sec^{2} \frac{k L}{2} & = & \frac{π^{2}}{2 L^{2}} \\ \cos^{2} \frac{k L}{2} & = & \frac{2 k^{2} L^{2}}{π^{2}} \end{array}$

This has a solution for

$k = \frac{π}{2 L}$

The corresponding energy is

$\begin{array}{rcl} E & = & \frac{ℏ^{2} k^{2}}{2 m} \\ = & \frac{ℏ^{2}}{2 m} \times \frac{π^{2}}{4 L^{2}} \\ = & \frac{π^{2} ℏ^{2}}{8 m L^{2}} \end{array}$

Thus the required energy is $\frac{π^{2} ℏ^{2}}{8 m L^{2}}$ .

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Short Answer

Step by step solution

Given data

Ground state energy of infinite potential well and height of the finite potential well

Determining the energy of the finite well

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