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Title & Summary Author Updated

kapoor 04/01/19

### PSU/QM-06-1 States and Dynamical Variables

Summary (or abstract) not available for this node.
kapoor 03/01/19

### PSU/QM-06-3 Compatiable Variables and Commuting Set

Summary (or abstract) not available for this node.
kapoor 03/01/19

### PPSU/QM-06-2 Probability and Average Values

Summary (or abstract) not available for this node.
kapoor 03/01/19

### PROBLEM/QM/10003

The spherical harmonics $Y_{lm}(\theta,\phi)$ are normalized simultaneous eigenfunctions of $L^2$ and $L_z$ operators. Use the co-ordinate space expressions \begin{eqnarray*} L_x &=& i\hbar \Big( \sin\phi {\partial\over \partial\theta } + \cot \theta
\cos\phi{\partial\over \partial \phi} \Big)\\ L_y &=& i\hbar \Big(-\cos\phi {\partial\over \partial\theta } + \cot \theta
\sin\phi{\partial\over \partial \phi} \Big)\\ L_z &=& -i\hbar {\partial\over\partial \phi} \end{eqnarray*}
\samepage{for the orbital angular momentum operators and the properties of thelladder\\operators, $L^\pm$, and construct expressions for $Y_{lm}(\theta,\phi)$ for $l=2$ and $m=2,1,0,-1,-2$.}

kapoor 03/01/19

### PROBLEM/QM-10004

1. Express the operators $a$ and $a^\dagger$ defined by $$a = {( p -i m\omega x) \over \sqrt{2m\omega \hbar}}, \qquad a^\dagger = {( p +i m\omega x) \over \sqrt{2m\omega \hbar}}$$ in the co-ordinate representation and solve the equation $$a \psi(x) = 0$$ to determine the ground state wave function.
2. Applying $a^\dagger$ on the ground state wave function, find the first two excited state eigen functions for the harmonic oscillator.
3. Normalize the ground state and the two excited state eigen functions found above.
kapoor 03/01/19

### PROBLEM/QM-10005

1. Find the matrices representing the operators $J_+, J_- \text{and} J_z$ in the basis $|jm\rangle.$
2. Use your answers in part (a) to find the matrices for the operators $J_x$ and $J_y$.
3. What answer do you expect for the matrix for $J^2$? Check if your guess is correct or not by computing the matrix for $L^2$ using the matrices found above.
kapoor 03/01/19

### PROBLEM/QM-10006

1. Express the following operators in terms of $a$ and $a^\dagger$. $$(i) \ \hat{x}\qquad (ii)\ \hat{p}\qquad (iii)\ \hat{x}^2 \qquad (iv)\ \hat{p}^2$$
2. Using the properties of operators $a$ and $a^\dagger$ compute the $m n$ matrix elements of the four operators given in part (a) in the harmonic oscillator basis.
3. What answer do you expect for the matrix elements of the Hamiltonian operator $$\hat{H} = { \hat{p}^2\over 2m} +{1\over2}m \omega^2 \hat{x}^2$$ Using the answers obtained in part (b) check if your guess is correct.
kapoor 03/01/19

### PROBLEM/QM-10007

Show that
• $\displaystyle \left[ \hat{p}_x, F(\vec{r}) \right] = -i \hbar {\partial F\over \partial x}$
• $\displaystyle \left[ \hat{x}, G(\vec{p}) \right] = -i \hbar {\partial G\over \partial p_x}$
kapoor 03/01/19

### PROBLEM/QM/10008

Compute  $$U(a)\ \vec{r} \ U^\dagger(a)$$  where  $$U(a) = \exp(-i\vec{a}\cdot \hat{\vec{p}})$$  where $\vec{a}$ are numbers.

kapoor 03/01/19