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[QUE/QM-08015]Node id: 2738page Assuming anticommutation relations \[ [a_m, a_n]_+=0, \quad [a_m, a_n^\dagger]_+=\delta_{mn}, [a_m^\dagger, a_n^\dagger]_+=0, \] Prove that the number operators \(N_n= a_n^\dagger a_n \) have eigenvalues 0 or 1 only. |
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22-04-11 17:04:33 |
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[QUE/QM-08014]Node id: 2739pageRearrange the operator expression \(pqpq^2\) as sum of expressions of the form \(\sum_{m,n}c_{mn}q_np_m\) in which each term has all \(q\) operators on the left and all \(p\) operators on the right. |
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22-04-11 17:04:20 |
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[QUE/QM-08013]Node id: 2740pageCompute average of kinetic and potential energies \[ \text{K.E.} = \frac{p^2}{2m},\qquad \text{P.E.}= \frac{1}{2}m\omega^2x^2.\] in the \(n^\text{th}\) excited state and show that the the average of total energy is \(\frac{1}{2}\hbar \omega\). |
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22-04-11 17:04:13 |
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[QUE/QM-08012]Node id: 2741pageUsing the properties of the ladder operators \(a, a^\dagger\) and the number operator \(N\), compute the average values of kinetic and potential energies for a harmonic oscillator in the \(n^\text{th}\) state \(|n\rangle\). Verify that their sum equals \((n + 1/2)\hbar\omega\). |
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22-04-11 17:04:54 |
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[QUE/QM-08011]Node id: 2742pageProve the following commutation relation \begin{equation} [L_z, (L_+)^n] = n \hbar (L_+)^n\end{equation} |
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22-04-11 17:04:18 |
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[QUE/QM-08010]Node id: 2743page Prove that if an operator commutes with any two of the three components of angular momentum, then it commutes with the third component also. |
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22-04-11 17:04:35 |
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[QUE/QM-08009]Node id: 2744pageCompute average of kinetic and potential energies \[ \text{K.E.} = \frac{p^2}{2m},\qquad \text{P.E.}= \frac{1}{2}m\omega^2x^2.\] in the \(n^\text{th}\) excited state and show that the the average of total energy is \(\frac{1}{2}\hbar \omega\). |
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22-04-11 13:04:52 |
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[QUE/QM-08008]Node id: 2745pageFor a harmonic oscillator in \(n^\text{th}\) excited state, compute \(\Delta q\) and \(\Delta p\) and show that \[ \Delta q \Delta p =\big(n+\frac{1}{2}\big)\hbar.\] |
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22-04-11 13:04:21 |
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[QUE/QM-08007]Node id: 2746pageLet $Y_{\ell m}(\theta,\phi)$ denote the simultaneous normalized eigenfunctions of $L^2$ and $L_z$ operators. Use the properties of the ladder operators, $L_\pm$, and construct the expressions for $Y_{lm}(\theta,\phi)$ for $l=2$ and $m=2,1,0,-1,-2$.
- Note that $Y_{\ell\ell}(\theta,\phi)$ satisfies\begin{eqnarray} L_z Y_{\ell\ell}(\theta,\phi) &=& \ell \hbar Y_{\ell\ell}(\theta, \phi)\nonumber\\ L_+ Y_{\ell\ell}(\theta,\phi) &=& 0. \nonumber\end{eqnarray} Set up the above differential equations and solve them using separation of variables and find (normalized) $Y_{2,2}$.
- Next apply $L_-$ repeatedly on \(Y_{2,2}\) and use $$L_- Y_{\ell, m} = \sqrt{\ell(\ell + 1)-m(m - 1)}\, \hbar\,Y_{\ell(m-1)}$$ to successively construct $Y_{2,m}$ for other values $m = 1, 0,-1,-2$.
- Normalize your answers and compare them with known expressions of $Y_{2m}(\theta,\phi), m=-2,-1,0,1,2.$
Hint: Use coordinate space representation for angular mormentum operators.
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22-04-11 13:04:17 |
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[QUE/QM-08006]Node id: 2747page
- What are the eigenvalues of $L^2 + \alpha L_x + \beta L_y + \gamma L_z$ for $\ell=2$. Give a full explanation for your answer.
- Construct matrices for $L_x,L_y, L_z$ for $\ell=1$ case and verify that $L^2$ is a multiple of identity.
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22-04-11 13:04:55 |
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[QUE/QM-08005]Node id: 2748pageThe the exact zero point energy of a system several uncoupled oscillators, all having frequency \(\omega_0\), is given to be \(314.5 \hbar \omega_0\). What is the energy of first excited state? |
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22-04-11 13:04:13 |
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[QUE/QM-08004]Node id: 2749pageLet \(|n\rangle\) denote the \(n^\text{th}\) excited state of a harmonic oscillator. Show that \begin{equation*} \langle n| x |m\rangle= \sqrt{\frac{\hbar}{2m\omega}}\Big( \sqrt{n+1}\delta_{m, n+1} + \sqrt{n}\delta_{m, n-1} \Big) \end{equation*} |
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22-04-11 13:04:41 |
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[QUE/QM-08003]Node id: 2750pageCompute average value \(\langle{n}|{q^4}|\rangle\) of \(q^4\) in the \(n^\text{th}\) energy state of harmonic oscillator. |
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22-04-11 13:04:13 |
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[QUE/QM-08002]Node id: 2751pageState if the combinations of $j,m$ values, in the table given below, are allowed or not. Complete the table by writing ALLOWED/NOT ALLOWED in the second column and specifying a reason in support of your answer selecting a reason from the list, $(R1)-(R5)$, given below. In case you do not find a valid or appropriate reason listed below, feel free to select option (R6) and specify your reason.\footnote{This question requires knowledge of angular momentum eigenvalues.}\\ {\bf{List of Possible Reasons:}}
- [(R1)]All values of $jm$ are allowed.
- [(R2)]Not allowed because $m$ is not an integer
- [(R3)]Allowed because all vales of $m$ in the range $-j$ to $+j$ are allowed
- [(R4)]Not allowed because $j,m$ must be an integers
- [(R5)]Not allowed because both $j,m$ must be integers, or half integers.
- [(R6)]Any other reason, please specify for each case separately
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22-04-11 13:04:33 |
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[QUE/QM-08001]Node id: 2752pageThe potential energy of two protons in hydrogen molecule ion in a model is given below \begin{eqnarray} V(x) &=& |E_1| f(x) \\ f(x) &=& - 1 + \frac{2}{x}\left[ \frac{(1-(2/3)x^2) e^{-x} + (1+x) e^{-2x}} {1+(1+x+x^2/3) e^{-x} }\right], \qquad x=R/a \end{eqnarray} $E_1= 13.6 \text{ eV}$ is the ground state energy of H atom and $a$ is the Bohr radius $\hbar^2/me^2$. The graph of this function $f(x)$ is reproduced below. Find numerical values of the bond length in ${A^o}$, the zero point energy and spacing of vibrational spectrum, both energies in electron volts.
NoteThe expression for $V(x)$ is taken from an approximate variational calculation of energy of the H molecule ion in Born Oppenheimer approximation. |
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22-04-11 13:04:49 |
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[QUE/QM-08001]Node id: 2753pageThe potential energy of two protons in hydrogen molecule ion in a model is given below
\begin{eqnarray} V(x) &=& |E_1| f(x) \\ f(x) &=& - 1 + \frac{2}{x}\left[ \frac{(1-(2/3)x^2) e^{-x} + (1+x) e^{-2x}} {1+(1+x+x^2/3) e^{-x} }\right], \qquad x=R/a \end{eqnarray} $E_1= 13.6 \text{ eV}$ is the ground state energy of H atom and $a$ is the Bohr radius $\hbar^2/me^2$. The graph of this function $f(x)$ is reproduced below. Find numerical values of the bond length in ${A^o}$, the zero point energy and spacing of vibrational spectrum, both energies in electron volts.
NoteThe expression for $V(x)$ is taken from an approximate variational calculation of energy of the H molecule ion in Born Oppenheimer approximation.
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22-04-06 16:04:34 |
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