Chapter 39: Q57P (page 1217)

An electron is trapped in a one-dimensional infinite potential well. Show that the energy difference $∆ E$ between its quantum levels n and n+2 is $(h^{2} / 2 m L^{2}) (n + 1)$ .

Short Answer

Expert verified

It is proved that $∆ E = (\frac{h^{2}}{2 m L^{2}}) (n + 1)$ .

Step by step solution

Describe the expression for Energy for the one-dimensional infinite potential well is given by

The Energy for the one-dimensional infinite potential well is given by,

$E_{n} = (\frac{h^{2}}{8 {mL}^{2}}) n^{2}$

Show that the energy difference ∆E is (h2/2mL2)(n+1)

Find the energy difference as follows.

$∆ E = E_{n + 2} - E_{n} = (\frac{h^{2}}{8 m L^{2}}) {(n + 2)}^{2} - (\frac{h^{2}}{8 m L^{2}}) n^{2} = (\frac{h^{2}}{8 m L^{2}}) [{(n + 2)}^{2} - n^{2}] = (\frac{h^{2}}{8 m L^{2}}) (n^{2} + 4 n + 4 - n^{2})$

Simplify further.

$∆ E = (\frac{h^{2}}{8 m L^{2}}) (n^{2} + 4 n + 4 - n^{2}) = (\frac{h^{2}}{8 m L^{2}}) (4) (n + 1) = (\frac{h^{2}}{8 m L^{2}}) (n + 1)$

Therefore, it is proved that $∆ E = (\frac{h^{2}}{8 m L^{2}}) (n + 1)$ .

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An electron is trapped in a one-dimensional infinite potential well. Show that the energy difference $∆ E$ between its quantum levels n and n+2 is $(h^{2} / 2 m L^{2}) (n + 1)$ .

Short Answer

Step by step solution

Describe the expression for Energy for the one-dimensional infinite potential well is given by

Show that the energy difference ∆E is (h2/2mL2)(n+1)

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An electron is trapped in a one-dimensional infinite potential well. Show that the energy difference∆E between its quantum levels n and n+2 is (h2/2mL2)(n+1).

Short Answer

Step by step solution

Describe the expression for Energy for the one-dimensional infinite potential well is given by

Show that the energy difference ∆E is (h2/2mL2)(n+1)

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Work Energy and Power

Physical Quantities And Units

Thermodynamics

Space Physics

Energy Physics

Kinematics Physics

Study anywhere. Anytime. Across all devices.

An electron is trapped in a one-dimensional infinite potential well. Show that the energy difference $∆ E$ between its quantum levels n and n+2 is $(h^{2} / 2 m L^{2}) (n + 1)$ .