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Title & Summary Author Updated

### For My Reference :: List of Sub topics for different Areas

Summary (or abstract) not available for this node.
kapoor 22/01/19

### Info-Page1 List of Modules

Summary (or abstract) not available for this node.
kapoor 22/01/19

### Help-Page 6 ---- Browse All Content

Summary (or abstract) not available for this node.
kapoor 22/01/19

### Help-Page 1 --- Quick Help on Navigation

Areas := Classical Mechanics, Quantum Mechanics  etc.
Categories := LectureNotes, Problems, Examples, Solved Problems, Courses, WebQuest and more ...

kapoor 22/01/19

### Problem/QM-07012

Consider the space of square integrable functions on a plane.  $\iint dx\,dy |\psi(x,y)|^2 < \infty.$  Define radial and angular momenta operators  $$\hat{p}_r, \hat{P}_\theta$$ on the subset of functions  satisfying  $\psi(r,\theta+2\pi) = \psi(r,\theta), \qquad \psi(r, \theta)|_{r=0} = \psi(r,\theta)|_{r\to \infty} =0 .$
1. Show that $${P}_{\theta}= -i\hbar\frac{\partial}{\partial\theta}$$ satisfies $\Big(\phi(r,\theta), \hat{P}_\theta \psi(r,\theta)\Big) = \Big(\hat{P}_\theta\phi(r,\theta), \psi(r,\theta)\Big)$ and is, therefore, a hermitian operator.
2. Find the hermitian conjugate of the operator $$\hat{p}_r\equiv-i\hbar\frac{\partial}{\partial r}$$. Show that $$\hat{p}_r$$ is not a hermitian operator.
3. Find a hermitian operator $$\hat{P}_r$$ that may represent radial momentum in two dimensions.
4. Consider the classical free Hamiltonian $$H_{cl} = \frac{P_r^2}{2m} + \frac{P_\theta^2}{2mr^2}.$$ Replace the classical momenta $$P_r, P_\theta$$ by corresponding hermitian momentum operators $$\hat{P}_r, \hat{P}_\theta$$. Compare your answer for the operator so obtained with the free particle Schr\"{o}dinger Hamiltonian $\widehat{H}_0 = -\frac{\hbar^2}{2m}\Big(\frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y}\Big).$ and give your comments.
kapoor 21/01/19

### Problem/QM-07013

Show that if an operator commutes with to components of angular momentum, it commutes with the third component as well.

{Daniel F. Styer}

kapoor 21/01/19

### QM-07 Canonical Quantization

This section has problems on canonical quantization and uncertainty relation.

kapoor 21/01/19

### Problem/QM-07014

Summary (or abstract) not available for this node.
kapoor 21/01/19

### Prob/QM-07015

{Let $$\ket{E_1},\ket{E_2}$$ denote normalized energy eigenstates with energies
$$E_1\ne E_2$$. Let $$\psi$$ be the superposition
$\ket{\psi} = a\ket{E_1} + b\ket{E_2},$
$$a,b$$ are complex constants. Obtain an expression for the uncertainty in
energy $$(\Delta E)_\psi$$in the state $$\ket{\psi}$$. Find all conditions
so that $$\Delta E$$ may be zero, and interpret the answers you get.}

kapoor 21/01/19

### QM-06 :: General Principles of Quantum Mechanics

Repository of problems on Postulates of Quantum Mechanics.
All problems fall  under "Analysis and Application" levels
of Bloom's Taxonomy.