Problem Set (due 10/17/02)

Exercise 16: (A) Show that, $<\Phi_{n'}\vert x\vert\Phi_n> = \sqrt{\frac{\hbar}{2m \omega}}[\sqrt{n+1}\delta_{n', n+1} + \sqrt{n}\delta_{n', n-1}]$.

(B) Show that, $<\Phi_{n'}\vert p\vert\Phi_n> = i \sqrt{\frac{m\hbar \omega}{2}}[\sqrt{n+1}\delta_{n', n+1} -\sqrt{n}\delta_{n', n-1}]$.

(C) Show that, $\hat{a}^+\vert\Phi_{\nu}> = \sqrt{\nu +1} \vert \Phi_{\nu +1}>;\ \hat{a}\vert\Phi_{\nu}> = \sqrt{n} \vert \Phi_{\nu -1}>$.

(D) Compute the ratio between the minimum vibrational energies for bonds C-H and C-D, assuming that the force constant $k=m\omega^2$ is the same for both bonds.

(E) Estimate the energy of the first excited vibrational state for a Morse oscillator defined as follows: $V(R)=D_e(1-$exp$(-a(R-R_{eq})))^2$.

Exercise 17: Prove that $<\Phi_k \vert\frac{\partial \hat{H}}{\partial t}\vert \Phi_n> = (E_n-E_k)<\Phi_k \vert\frac{\partial} {\partial t}\vert \Phi_n>$, when $n \neq k$ and $<\Phi_k \vert \Phi_n> =\delta_n$, with

$$\hat{H}(t) \Phi_j(x,t)= E_j(t) \Phi_j(x,t).$$

Exercise 18: Prove that $\nabla \cdot j =0,$ where $j \equiv \frac{\hbar}{2mi} ( \psi^* \frac{\partial \psi}{\partial x}-\psi \frac{\partial \psi^*}{\partial x})$ and $\psi= R(x) e^{-\frac{i}{\hbar}Et}.$

Exercise 19: Consider a harmonic oscillator described by the following Hamiltonian,

$$\hat{H}_0 = \frac{1}{2m} p^2 + \frac{1}{2} m \omega^2 x^2.$$

Consider that the system is initially in the ground state $\Phi_0$, with

$$\hat{H}_0 \Phi_k =E_k \Phi_k,$$   with$$\qquad E_k= \hbar \omega(\frac{1}{2} +k).$$

Compute the probability of finding the system in state $\Phi_2$ at time t after suddenly changing the frequency of the oscillator to $\omega '$.