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

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kapoor kapoor's picture 22/01/19

Info-Page1 List of Modules

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kapoor kapoor's picture 22/01/19

Help-Page 6 ---- Browse All Content

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Help-Page 1 --- Quick Help on Navigation

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

kapoor kapoor's picture 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 kapoor's picture 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 kapoor's picture 21/01/19

QM-07 Canonical Quantization

This section has problems on canonical quantization and uncertainty relation.

kapoor kapoor's picture 21/01/19

Problem/QM-07014

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kapoor kapoor's picture 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 kapoor's picture 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. 

 

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kapoor kapoor's picture 21/01/19

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