Chapter 5: Bound States: Simple Cases

The product of uncertainties in particle's momentum and position.

Determine the expectation value of the position of a harmonic oscillator in its ground state.

Show that the uncertainty in the position of a ground state harmonic oscillator is $\left(\mathbf{1}\mathbf{/}\sqrt{\mathbf{2}}\right){\left({\hslash }^{\mathbf{2}}\mathbf{/}\mathbf{mk}\right)}^{\mathbf{1}\mathbf{/}\mathbf{4}}$.

Show that the uncertainty in the momentum of a ground state harmonic oscillator is $\left(\frac{\hslash }{\sqrt{\mathbf{2}}}\right){\left(\mathbf{mk}\right)}^{\mathbf{1}\mathbf{/}\mathbf{4}}$.

What is the product of uncertainties determined in Exercise 60 and 61? Explain.

Repeat the exercise 60-62 for the first excited state of harmonic oscillator.

If a particle in a stationary state is bound, the expectation value of its momentum must be 0.

(a). In words, why?

(b) Prove it.

Starting from the general expression(5-31) with $\hat{\mathbf{p}}$in the place of $\overbrace{\mathbf{Q}}$, integrate by parts, then argue that the result is identically 0. Be careful that your argument is somehow based on the particle being bound: a free particle certainly may have a non zero momentum. (Note: Without loss of generality,$\mathit{\psi }\mathbf{\left(}\mathit{x}\mathbf{\right)}$ may be chosen to be real.)

equation (5-33). The twosolutionsare added in equal amounts. Show that if we instead added a different percentage of the two solutions. It would not change the important conclusion related to the oscillation frequency of the charge density.

Consider the wave function that is a combination of two different infinite well stationary states the $n^{th}$ and the $m^{th}$

$\psi \left(x,t\right)=\frac{1}{\sqrt{2}}{\psi }_n\left(x\right)e^{-i\left(E_n/\hslash \right)t}+\frac{1}{\sqrt{2}}{\psi }_m{e}^{-i\left(E_m/\hslash \right)t}$

1. Show that the $\psi \left(x,t\right)$is properly normalized.
2. Show that the expectation value of the energy is the average of the two energies:$\overline{E}=\frac{1}{2}\left(E_n+E_m\right)$
3. Show that the expectation value of the square of the energy is given by .
4. Determine the uncertainty in the energy.

Prove that the transitional-state wave function (5.33) does not have a well-defined energy.