Chapter 5: Q36E (page 188)

Whereas an infinite well has an infinite number of bound states, a finite well does not. By relating the well height $U_{0}$ to the kinetic energy and the kinetic energy (through $λ$ ) to n and L. Show that the number of bound states is given roughly by $\sqrt{8 m l^{2} U_{0} / h^{2}}$ (Assume that the number is large.)

Short Answer

Expert verified

Hence, the number of bound states is given by $n_{\max} \approx \sqrt{\frac{8 U_{0} m L^{2}}{h^{2}}}$ is proved

Step by step solution

Assumption

We assume that the wave function and energies correspond to the infinite well.

If, $E < U_{0}$ then inside the well, the energy is totally kinetic and is given by,

$\frac{h^{2} k^{2}}{2 m} = \frac{n^{2} π^{2} h^{2}}{2 m L^{2}}$

Calculation

The highest bound state exists when $E = U_{0}$ . i.e., the difference $E - U_{0} \to 0$ . For this state,

$\begin{matrix} \frac{n_{\max}^{2} π^{2} h^{2}}{2 m L^{2}} \approx U_{0} \\ n_{\max} \approx \sqrt{\frac{8 U_{0} m L^{2}}{h^{2}}} \end{matrix}$

Hence the number of bounded states $n_{\max} \approx \sqrt{\frac{8 U_{0} m L^{2}}{h^{2}}}$ is proved.

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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 λ) to n and L. Show that the number of bound states is given roughly by8ml2U0/h2 (Assume that the number is large.)

Short Answer

Step by step solution

Assumption

Calculation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Medical Physics

Work Energy and Power

Linear Momentum

Physical Quantities And Units

Further Mechanics and Thermal Physics

Magnetism

Study anywhere. Anytime. Across all devices.