Chapter 11, Angular Momentum in Quantum Mechanics Video Solutions, Quantum mechanics

Problem 1

For the state represented by the wave function
$$
\psi=N e^{-\alpha r^2}(x+y) z
$$
(a) Determine the normalization constant $N$ as a function of the parameter $\alpha$.
(b) Calculate the expectation values of $\mathbf{L}$ and $L^2$.
(c) Calculate the variances of these quantities.

Susan Hallstrom

Numerade Educator

04:56

Problem 2

. For a finite rotation by an angle $\alpha$ about the $z$ axis, apply the rotation operator $U_R$ to the function $f(r)=a x+b y$, and show that it transforms correctly.

Adriano Chikande

Numerade Educator

Problem 3

Explicitly work out the $\mathbf{J}$ matrices for $j=1 / 2,1$, and $3 / 2$.

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Problem 4

Classically, we have for central forces
$$
H=\frac{p_r^2}{2 m}+\frac{\mathbf{L}^2}{2 m r^2}+V(r)
$$
where $p_r=(1 / r)(\mathbf{r} \cdot \mathbf{p})$. Show that for translation into quantum mechanics we must write
$$
p_r=\frac{1}{2}\left[\frac{1}{r}(\mathbf{r} \cdot \mathbf{p})+(\mathbf{p} \cdot \mathbf{r}) \frac{1}{r}\right]
$$
and that this gives the correct Schrödinger equation with the Hermitian operator
$$
p_r=\frac{\hbar}{i}\left(\frac{\partial}{\partial r}+\frac{1}{r}\right)
$$
whereas $(\hbar / i)(\partial / \partial r)$ is not Hermitian.

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Problem 5

Show that in $D$-dimensional Euclidean space the result of Problem 4 generalizes to
$$
p_r=\frac{1}{2}\left[\frac{1}{r}(\mathbf{r} \cdot \mathbf{p})+(\mathbf{p} \cdot \mathbf{r}) \frac{1}{r}\right]=\frac{\hbar}{i}\left(\frac{\partial}{\partial r}+\frac{D-1}{2 r}\right)
$$

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