Chapter 40, Problem 40.64CP

(a)

To determine

To show: The wave length of a quantum mechanical particle in a potential well $U\left(x\right)$ as being like a free particle as a function of position $x$ can be written as $\lambda \left(x\right)=\frac{h}{\sqrt{2m\left[E-U\left(x\right)\right]}}$.

(b)

To determine

What happens to its wavelength if the particle moves into the region of increasing potential energy.

(b)

To determine

What happens to its wavelength if the particle moves into the region of increasing potential energy.

(c)

To determine

The wavelength at classical turning point.

(d)

To determine

To show: That the WKB condition for an allowed bound state energy can be written as ${\displaystyle {\int }_a^b\sqrt{2m\left[E-U\left(x\right)\right]}dx}=\frac{nh}{2}$ for $n=1,2,3,\mathrm{....}$.

(e)

To determine

To show: That Energy for particle in a box with infinite potential walls at and $x=L$ using WKB approximation is equal to $\frac{n^2{h}^2}{8m{L}^2}$, where $n=1,2,\mathrm{..}$.

(f)

To determine

The integral ${\displaystyle {\int }_a^b\sqrt{2m\left[E-U\left(x\right)\right]}dx}=\frac{nh}{2}$ for a finite potential well with walls at $x=0$ and $x=L$ then show that allowed energy levels according to the WKB approximation are the same as those obtained in infinite potential well.