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[QUE/QM-16002]Node id: 2598page
Use the following data on the wavelengths of the first few lines of the Lyman series of the hydrogen atom to find best fit to the value of the Rydberg constant.
Wavelengths in Angstrom units :: 1215.68, 1025.83, 972.54
Use your value of the Rydberg constant to predict the wavelengths of the lines
specified below.
(i) next two lines in the Lyman series.
(ii) the first nine lines in the Balmer series.
(iii) the first three lines of the Ritz Paschen series.
(iv) the first two lines of the Brackett series.
(v) Compare your answers with the experimental results given at the
end and comment on the agreement of the theory with the experiments.
Remarks:
Experimental wavelengths are as follows
(i) Balmer series : 6562.79,
4861.33, 4340.17, 4104.74, 3970.07, 3889.05, 3835.39, 3797.90, 3770.63 , all in
Angstrom units.
(ii) Ritz Paschen Series : 18751.1, 12818.1, 10938 , all in Angstrom units.
(iii) Brackett Series : 2.63 $\mu$, 4.05 $\mu$
Latest value of the Rydberg constant, R = 109737.3177 $\mbox{cm}^{-1}$
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22-04-12 09:04:06 |
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[QUE/QM-16004]Node id: 2599page
- For a classical electron moving in an orbit around a nucleus with charge $Ze$ obtain a relation between the number of revolutions per second and the energy of the electron. (Use only classical mechanics).
- Using known quantum mechanical solution, show that in the limit of large principal quantum number \(n\), the frequency of the line emitted in a transition from $(n+1)^{\mbox{th}}$ level to the $n^{\mbox{th}}$ level is precisely the same as the frequuency obtained in part \((a)\).
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22-04-12 09:04:01 |
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[QUE/QM-16009]Node id: 2820page
- Show that the average value of kinetic energy for a particle in one dimension having the wave function \(\psi(x)\) is \[ \langle \text{K.E.}\rangle = \frac{\hbar^2}{2m}\int_{-\infty} ^ \infty |\psi(x)|^2\, dx.\]
- Obtain a similar formula for the average of kinetic energy in three dimensions for a particle if the wave function \(\psi(r)\) is independent of polar coordinates \(\theta, \phi\).
- Can one write down a similar result for the general case in polar coordinates when the wave fucntion depends on all the three variables \(r,\theta,\phi\)?
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22-04-10 18:04:04 |
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[QUE/QM-16008]Node id: 2821page
- Express the angular momentum operators \begin{eqnarray} \hat{L}_x &=& -i\hbar{\hat{y} \frac{\partial}{\partial z} - \hat{z} \frac{\partial}{\partial y} }\label{EQ01}\\ \hat{L}_y &=& -i\hbar{\hat{z} \frac{\partial}{\partial x} - \hat{x} \frac{\partial}{\partial z} }\label{EQ02}\\ \hat{L}_z &=& -i\hbar{\hat{x} \frac{\partial}{\partial y} - \hat{y} \frac{\partial}{\partial x} }\label{EQ03} \end{eqnarray} in polar coordinates and show \begin{eqnarray} \hat{L}_x &=& i\hbar \left(\sin\phi \frac{\partial}{\partial \theta} +\cot\theta\cos\phi\frac{\partial}{\partial \phi} \right)\label{EQ04}\\ \hat{L}_x &=& i\hbar \left(\sin\phi \frac{\partial}{\partial \theta} +\cot\theta\cos\phi\frac{\partial}{\partial \phi} \right)\label{EQ05}\\ \hat{L}_z &= & i\hbar \frac{\partial }{\partial \phi}\label{EQ06} \end{eqnarray}
- Use the result of the previous part and show that The operator $\vec{L}^2$ in spherical polar coordinates is given by \begin{equation} \vec{L}^2 = \hat{L}_x^2 +\hat{L}_y^2 + \hat{L}_z^2 \label{EQ07} \end{equation} takes the form \begin{equation}\label{EQ08} \vec{L}^2 = -\hbar^2\left[ \frac{1}{\sin\theta} \frac{\partial}{\partial\theta}\left(\sin\theta \frac{\partial}{\partial\theta} \right) + \frac{1}{\sin^2\theta}\frac{\partial^2}{\partial \phi^2} \right] \end{equation}
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22-04-10 18:04:26 |
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[QUE/QM-16010]Node id: 2819pageHow much energy in eV will be required to ionize a H atom which is in the excited state \(n=3\)? Use calculator to get the answer upto significant number of places. Take the value of the Rydberg constant to be \(R = 109737.3177 \mbox{cm}^{-1}\) |
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22-04-10 11:04:07 |
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[QUE/QM-16011]Node id: 2818pageThe wave function of a particle has the form \begin{equation*} \psi(r\theta,\phi) = \chi(r)( A\cos^7\theta + B \cos^3\theta + C \cos2\phi) \end{equation*} where \(\chi(r)\) is radial part of the wave function dependent only on \(r\). A measurement of \(\vec{L}^2\) and \(L_z\) is made. What values do you expect for \(\vec{L}^2\)? for \(L_z\)? |
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22-04-10 11:04:01 |
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[QUE/QM-16001]Node id: 2827pageFor each of the six potentials, shown in the \Figref{3dpot1} below, answer thefollowing.
- \label{16001Q1} Write the radial Schrodinger equation in different regions of $r$ values.
- Write the most general solution as a linear combinations of spherical Bessel functions with appropriate arugments.
- Impose the necessary regularity property at the origin and at $\infty$.
- Impose the matching conditions at the boundary of two neighbouring regions.
- Considering appropriate ranges of energy separately, state if energy eigenvalues are continuous or discrete?
- \label{16001Q2} Whenever bound states exist for a potential, obtain the condition on the bound state energy eigenvalues.
\FigBelow{0,0}{150}{220}{3dpot1}{Radial Wells for Q.
\ref{16001Q1}-\ref{16001Q2}}
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22-04-10 11:04:31 |
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[QUE/QM-16005]Node id: 2824pageClassically orbit of a particle in a spherically symmetric potential lies in a plane. Show that the classical result is approximated by quantum solution infor \(L_z=\ell\) the limit of large angular momentum \(\ell\). |
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22-04-10 11:04:20 |
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[QUE/QM-16003]Node id: 2826pageWhat qualitative differences do you expect in the spectra of the
H- atom and
(i) singly ionized Helium atom\\
(ii) the deuterium
(iii) positronium |
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22-04-10 11:04:56 |
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[QUE/QM-16004]Node id: 2825page
- For a classical electron moving in an orbit around a nucleus with charge $Ze$ obtain a relation between the number of revolutions per second and the energy of the electron. (Use only classical mechanics).
- Using known quantum mechanical solution, show that in the limit of large principal quantum number \(n\), the frequency of the line emitted in a transition from $(n+1)^{\mbox{th}}$ level to the $n^{\mbox{th}}$ level is precisely the same as the frequuency obtained in part \((a)\).
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22-04-10 11:04:06 |
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[QUE/QM-16006]Node id: 2823pageClassically orbit of a particle in a spherically symmetric potential lies in a plane. Show that the classical result is approximated by quantum solution in for \(L_z=\ell\) the limit of large angular momentum \(\ell\). |
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22-04-09 17:04:10 |
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[QUE/QM-16007]Node id: 2822pagethe following statements correctly describes the degeneracy?\\
- Eigen values are non-degenerate just as for the bound states in one dimension.
- Continuous eignevalues are doubly degenerate as in some one dimensional problems.
- Continuous eignevalues are non-degenerate as in some one dimensional problems.
- Infinitely degenerate as is the case for free particle in three dimension.
- \label{16007Q2} The degeneracy is $(2l+1)$ as expected for a general spherically symmetric potential in three dimensions.
\FigBelow{0,0}{150}{200}{3dpot2}{Radial Wells for Q. |
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22-04-09 17:04:30 |
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