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Exploring quantum mechanics: a collection of 700+ solved problems for students, lecturers, and researchers

Victor Galitski Jr., Vladimir Kogan, Boris Karnakov

Chapter 3

Orbital angular momentum - all with Video Answers

Educators

Chapter Questions

03:04

Problem 1

Show that the relation $\mathbf{L}^2=l(l+1)$ could be obtained by using elementary equations of probability theory. Assume that the only possible values of angular momentum projection on an arbitrary axis are $m=-l,-l+1, \ldots, l$, and that all of these values have equal probability and all the axes are equivalent.

Keshav Singh
Keshav Singh
Numerade Educator
03:14

Problem 2

Find the stationary wavefunctions and energy levels of a planar (two-dimensional) rotor $^{[25]}$ with a moment of inertia $I$. What is the degeneracy multiplicity?
Find the probabilities for different energy and angular momentum projection values, as well as the mean values and fluctuations of these quantities, for the rotor with the wavefunction $\psi=C \cos ^2 \varphi$.

Lottie Adams
Lottie Adams
Numerade Educator
03:14

Problem 3

Find the wavefunctions and energy levels of the stationary states of a spherical (threedimensional) rotor with a momentum of inertia $I$. What is the degeneracy multiplicity?
Let the wavefunction be $\psi=C \cos ^2 \vartheta$. Find the probability distribution for energy, angular momentum, and $z$-axis angular momentum. Find the mean values and fluctuations of these quantities.

Lottie Adams
Lottie Adams
Numerade Educator
01:59

Problem 4

Give a simple explanation of
a) the commutativity of different components of the momentum operator;
b) the non-commutativity of the angular momentum components;
c) the commutativity of the momentum projection and angular momentum projection on the same axis, and their non-commutativity for different axes, using kinematic interpretation of these operators in terms of infinitely small translations and rotations.

Dheeraj Sharma
Dheeraj Sharma
Numerade Educator
04:14

Problem 5

Find the following commutators:
a) $\left[\hat{l}_i, \hat{\mathbf{r}}^2\right],\left[\hat{l}_i, \hat{\mathbf{p}}^2\right],\left[\hat{l}_i, \hat{\mathbf{p}} \cdot \hat{\mathbf{r}}\right],\left[\hat{l}_i,(\hat{\mathbf{p}} \cdot \hat{\mathbf{r}})^2\right]$;
b) $\left[\hat{l}_i,(\hat{\mathbf{p}} \cdot \hat{\mathbf{r}}) \hat{p}_k\right],\left[\hat{l}_i,(\hat{\mathbf{p}} \cdot \hat{\mathbf{r}}) \hat{x}_k\right],\left[\hat{l}_i,\left(c_1 \hat{x}_k+c_2 \hat{p}_k\right)\right]$;
c) $\left[\hat{l}_i, \hat{x}_k \hat{x}_l\right],\left[\hat{l}_i, \hat{p}_k \hat{p}_l\right],\left[\hat{l}_i, \hat{x}_k \hat{p}_l\right]$.
Here, $c_1, c_2$ are some constants.

Andrija Isakov
Andrija Isakov
Numerade Educator
17:30

Problem 6

Find the normalized wavefunctions $\psi_{r_0 l m}$ that describe a particle located at a distance $r_0$ from the origin with angular momentum $l$ and its projection $m$ onto the axis $z$.

Yaqub Khan
Yaqub Khan
Numerade Educator
01:12

Problem 7

Find general eigenfunctions of particle momentum and angular momentum projections on the $z$ axis.

Narayan Hari
Narayan Hari
Numerade Educator
02:03

Problem 8

Show that the mean values of the vectors $\overline{\mathbf{L}}, \overline{\mathbf{r}}, \overline{\mathbf{p}}$ for the particle state with wavefunction $\psi=\exp \left(i \mathbf{p}_0 \cdot \mathbf{r} / \hbar\right) \varphi(\mathbf{r})$ are connected by the classical relation $\overline{\mathbf{L}}=\overline{\mathbf{r}} \times \overline{\mathbf{p}}$. Here, $\mathbf{p}_0$ is a real vector and $\varphi(\mathbf{r})$ is a real function.

Lottie Adams
Lottie Adams
Numerade Educator
01:16

Problem 9

Find the eigenfunctions of the operators $\widehat{\mathbf{l}}^2$ and $\hat{l}_z$ in the momentum representation. Show that $\overline{\mathbf{p}}=0$ for the states with definite values of $l$ and $m$.

Lottie Adams
Lottie Adams
Numerade Educator
05:56

Problem 10

Prove that the functions produced by the action of operators $\hat{l}_{ \pm}=\hat{l}_x \pm i \hat{l}_y$ on the eigenfunctions $\psi_m$ of $\hat{l}_z$, are also the eigenfunctions of $\hat{l}_z$, corresponding to eigenvalues $m \pm 1$
Show also that for eigenfunctions of $\hat{l}_z$, we have
a) $\hat{l}_x=\hat{l}_y=0$
b) $\hat{l}_x^2=\hat{l}_y^2$
c) $\overline{\hat{l}_x \hat{l}_y+\hat{l}_y \hat{l}_x}=0$

Abhijit Das
Abhijit Das
Numerade Educator
02:28

Problem 11

In the state $\psi_{l m}$ with definite angular momentum $l$ and its $z$-component $m$, find the mean values $\bar{l}_x^2, \overline{l_y^2}$ as well as the mean values $\bar{l}_{\tilde{z}}$ and $\overline{l_{\tilde{z}}^2}$ of the angular momentum projection along the $\tilde{z}$-axis making an angle $\alpha$ with the $\tilde{z}$-axis.

Dheeraj Sharma
Dheeraj Sharma
Numerade Educator

Problem 12

Prove the relation
$$
\sum_{m=-l}^l\left|Y_{l m}(\vartheta, \varphi)\right|^2=\frac{2 l+1}{4 \pi} .
$$

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Problem 13

Find the form of wavefunction $\psi_{l, \tilde{m}=0}(\mathbf{n})$ of a particle with total angular momentum $l$ and projection along the $\tilde{z}$-axis $\tilde{m}=0$. In this state, determine the probabilities of different values for the $z$-component of the angular momentum.

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03:14

Problem 14

Let $w_l\left(m_1 ; m_2, \alpha\right)$ be the probability to measure a particle's projection of the angular momentum on the $\tilde{z}$-axis as $m_2$, if the particle is in the state with a definite angular momentum, $m_1$, along the $z$ axis, where both states have definite angular momentum $l$ and the angle between the axes is $\alpha$. Prove that $w_l\left(m_1 ; m_2, \alpha\right)=w_l\left(m_2 ; m_1, \alpha\right)$.

Manish Jain
Manish Jain
Numerade Educator

Problem 15

For an angular momentum $L$, find the projection operators $\hat{P}_L(M)$ that project onto the state with a definite value $M$ for its $z$-component. The operators act in the space of the state vectors with a given value $L$ of the angular momentum.

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Problem 16

Using only the commutation relations for the components of the angular momentum, find $\operatorname{Tr}\left(\hat{l}_i\right)$, where $\hat{l}_i$ is a matrix of the $i$ th component of angular momentum $l$.

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05:56

Problem 17

Determine the traces of the following matrices:
a) $\hat{L}_i$,
b) $\hat{L}_i \hat{L}_k$,
c) $\hat{L}_i \hat{L}_k \hat{L}_l$,
d) $\hat{L}_i \hat{L}_k \hat{L}_l \hat{L}_m$,
where $\hat{L}_i$ is a matrix of the $i$ th component of angular momentum $L$.

Lucas Finney
Lucas Finney
Numerade Educator
17:30

Problem 18

For the case of a particle with the angular momentum $l=1$, find the wavefunction $\psi_{\tilde{m}=0}(\vartheta, \varphi)$ of the state with a definite projection $\tilde{m}=0$ of the angular momentum on the $\tilde{z}$-axis whose polar and azimuthal angles are $\alpha$ and $\beta$.

Yaqub Khan
Yaqub Khan
Numerade Educator

Problem 19

Find the wavefunctions $\psi_{l x}(\vartheta, \varphi)$ and $\psi_{l y}(\vartheta, \varphi)$ of a particle having a given value $l=1$ of angular momentum and a definite value of its projection onto $x$ and $y$ axes. Use the specific form of the spherical harmonics $Y_{1 m}(\vartheta, \varphi)$. See Eq. (III.7).

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02:19

Problem 20

A particle is in a state with angular momentum $l=1$ and $z$-projection $m(m=0, \pm 1)$. For such a state, determine the probabilities, $w(\tilde{m}, m)$, of different values $\tilde{m}$ of the angular-momentum projections onto the $\tilde{z}$-axis making an angle $\alpha$ with $z$-axis.
You can use one of the following two approaches to the problem:
a) by using the result of Problem 3.11 ;
b) by finding the expansion coefficient $c(\tilde{m}, m)$ of the given wavefunction into a series of eigenfunctions of the operator $\hat{l}_{\tilde{z}}$.

Dominador Tan
Dominador Tan
Numerade Educator

Problem 21

Show that for a particle with angular momentum $l=1$, the three functions $\psi_{l_x=0}(\vartheta, \varphi), \psi_{l_y=0}(\vartheta, \varphi)$, and $\psi_{l_z=0}(\vartheta, \varphi)$, that correspond to the states where the projection of the angular momentum onto the $x-, y$-, and $z$-axis correspondingly is zero, form a complete set of functions.
What is the meaning of the expansion coefficients of an arbitrary state with $l=1$ in terms of these functions?

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01:32

Problem 22

For the angular momentum $l=1$, write expressions for the operators of angular momentum components, as well as for raising $\hat{l}_{+}$and lowering $\hat{l}_{-}$operators, in the $l_z$-representation.
Find the wavefunction of a state with $l_x=0$ in the $l_z$-representation from the solution of an eigenfunction equation.

Jayashree Behera
Jayashree Behera
Numerade Educator

Problem 23

For a state with the value of angular momentum $l=1$ and its z-projection $m$, find the mean values $\bar{l}_x^n$ and $\bar{l}_y^n$ ( $n$ is integer).

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19:29

Problem 24

Find an explicit form of the operator $\hat{R}\left(\varphi_0\right)=\exp \left(i \varphi_0 \cdot \hat{\mathbf{l}}\right.$ ) (a coordinate system rotation over the angle, $\left.\phi_0\right)$ that acts in the space of state vectors with angular momentum $l=1$. Using this operator, obtain in terms of the spherical function $Y_{10}$, the wavefunction, $\psi_{\tilde{m}=0}(\vartheta, \varphi)$, of a state with $l=1$ and $\tilde{m} \equiv l_{\tilde{z}}=0$, where the $\tilde{z}$-axis is defined by its polar $\alpha$ and azimuth $\beta$ angles. Compare with Problem 3.18.

Hafiz Shahzaib
Hafiz Shahzaib
Numerade Educator

Problem 25

In the space of states with angular momentum $l=1$, find the projection operators, $\hat{P}(m)$, to states with a definite $z$-component of the angular momentum, $m$.
Generalize the results obtained to the case of an arbitrarily directed $\tilde{z}$-axis. By using the operator $\hat{P}_{\tilde{m}}$, obtain both in the $l_z$ and in the coordinate representations the wavefunction, $\psi_{1, \tilde{m}=0}$, of a state with angular momentum $l=1$ and $\tilde{z}$-projection $\tilde{m}=0$. Compare with Problem 3.18 and Problem 3.24.

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01:29

Problem 26

Write down the total angular momentum operator of two particles as a sum of two terms, corresponding to the angular momentum in the center of inertia system (i.e., the angular momentum of relative motion) and the angular momentum in the frame of reference associated with the system's translational motion as a whole.

Adriano Chikande
Adriano Chikande
Numerade Educator

Problem 27

Angular momenta $l_1$ and $l_2$ of two weakly interacting systems are combined into a resulting angular momentum with the value $L$. Show that in such states (with a definite $L)$ the products $\hat{\mathbf{l}}_1 \cdot \hat{\mathbf{l}}_2, \hat{\mathbf{l}}_1 \cdot \hat{\mathbf{L}}, \hat{\mathbf{l}}_2 \cdot \hat{\mathbf{L}}$ have definite values as well.

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14:42

Problem 28

Find the following commutators:
1) $\left[\hat{L}_i,\left(\hat{\mathbf{l}}_1 \cdot \hat{\mathbf{l}}_2\right)\right],\left[\hat{L}_i,\left(\hat{\mathbf{r}}_1 \cdot \hat{\mathbf{p}}_2\right)\right],\left[\hat{L}_i,\left(\hat{\mathbf{r}}_1 \cdot \hat{\mathbf{r}}_2\right)\right]$
2) $\left[\hat{L}_i, \hat{x}_{1 k}\right],\left[\hat{L}_i, \hat{g}_k\right]$ where $\hat{\mathbf{g}}=\left[\hat{\mathbf{l}}_1 \times \hat{\mathbf{l}}_2\right]$;
3) $\left[\hat{L}_i, \hat{x}_{1 k} \hat{x}_{2 l}\right],\left[\hat{L}_i, \hat{x}_{1 k} \hat{p}_{2 l}\right]$,
$\hat{\mathbf{l}}_1$ and $\hat{\mathbf{l}}_2$ are the angular momentum operators of particles 1 and $2, \hat{\mathbf{L}}=\hat{\mathbf{l}}_1+\hat{\mathbf{l}}_2$ is the operator of their total angular momentum. Note that the commutators have a universal structure (inside each group of expressions presented above). Compare with Problem 3.5.

Mahnoor Amin
Mahnoor Amin
Numerade Educator
03:04

Problem 30

Show that when we add two angular momenta of the same value $\left(l_1=l_2=l\right)$ to produce a total angular momentum, $L$, the wavefunction $\psi_L\left(m_1, m_2\right)$ in the $l_{1 z} l_{2 z^{-}}$ representation has a symmetry with respect to interchange of $m_1$ and $m_2$. Indicate how the nature of the symmetry depends on the value of $L$.

Keshav Singh
Keshav Singh
Numerade Educator
03:04

Problem 31

A system with the $z$-projection of the angular momentum $M$ is made up of two particles with the same total angular momentum values, $l_1=l_2$. Prove that the probabilities of $m_{1(2)}=m$ and $m_{1(2)}=M-m$ are the same.

Keshav Singh
Keshav Singh
Numerade Educator
02:19

Problem 32

Two subsystems which have the same values of their angular momenta, $l_1=l_2=1$, are in states with definite values $m_1$ and $m_2$ of the angular momentum projections. Determine the probabilities for different values, $L$, of the total angular momentum in such states. Use the result of Problem 3.29 for the value of $\bar{L}^2$ and take into account the symmetry of the state wavefunction with a definite value of $L$ shown in Problem 3.30. We should note that with arbitrary values of $l_{1,2}$ and $m_{1,2}$, the desired probability is $w(L)=\left|C_{l_1 m_1 l_2 m_2}^{L, m_1+m_2}\right|^2$, where $C_{l_1 m_1 l_2 m_2}^{L M}$ are the Clebsch-Gordan coefficients. See Problem 3.38.

Dominador Tan
Dominador Tan
Numerade Educator

Problem 33

Illustrate the relation established in Problem 1.43 and its probabilistic interpretation by the example of the addition of the angular momenta $l_1$ and $l_2$ for two weakly interacting subsystems with the total angular momentum, $L$.

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Problem 34

For a system of two particles with equal angular momenta $l_1=l_2=l$, find the wavefunction of a state with $L=0$ in the $l_{1 z} l_{2 z}$-representation. Use the operators $\hat{L}_{ \pm}$. Find the wavefunction in the coordinate representation also.

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02:19

Problem 35

The angular momenta of two particles are $l_1=l_2=1$. For such a system, find the wavefunctions $\psi_{L M}$ of states with given values $L$ and $M$ of the total angular momentum and its $z$-projection. Use the results of Problems 3.30 and 3.34.

Dominador Tan
Dominador Tan
Numerade Educator

Problem 36

For a system of two angular momenta, $l_1=l_2=1$, find the wavefunction, $\psi_{L=0}$, with $L=0$ total angular momentum, using the projection operators. Compare with Problem 3.34.

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Problem 37

Classify the independent states of a system which consists of three weakly interacting subsystems whose angular momenta are $l_1=l_2=1$ and $l_3=l$, by the value of the total angular momentum $L$.

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Problem 38

As is known, the problem of the addition of angular momenta of two systems $l_1$ and $l_2$ into the total angular momentum $L$ could be solved by the following relation
$$
\psi_{L M}=\sum_{m_1 m_2} C_{l_1 m_1 l_2 m_2}^{L M} \psi_{l_1 m_1}^{(1)} \psi_{l_2 m_2}^{(2)}, M=m_1+m_2,
$$where $C_{l_1 m_1 l_2 m_2}^{L M}$ are the Clebsch-Gordan coefficients. Using the "raising" ("lowering")-operators, $\hat{L}_{ \pm}$, determine these coefficients for the special case of $L=l_1+l_2$.

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13:54

Problem 39

$$
\text { The same as for the previous problem but for the case where } l_1=l_2, L=0 \text {. }
$$

Ahmad Reda
Ahmad Reda
Numerade Educator
01:31

Problem 40

For two weakly interacting systems with $j_1$ and $j_2$ for their angular momenta, average the operators
a) $\hat{j}_{1(2) i}$,
b) $\hat{j}_{1 i} \hat{j}_{2 k}-\hat{j}_{1 k} \hat{j}_{2 i}$;
c) $\hat{j}_{1 i} \hat{j}_{2 k}+\hat{j}_{1 k} \hat{j}_{2 i}$
d) $\hat{j}_{1 i} \hat{j}_{1 k}+\hat{j}_{1 k} \hat{j}_{1 i}$
over the states characterized by a given value, $J$, of the total angular momentum.
Obtain an explicit expression for the magnetic moment operator of the system, $\hat{\boldsymbol{\mu}}=g_1 \hat{\mathbf{j}}_1+g_2 \hat{\mathbf{j}}_2$, in a state with a given total angular momentum, $J$ (here $g_{1,2}$ are the gyro-magnetic ratios for the subsystems considered that relates their magnetic and mechanic angular momenta).

Suzanne W.
Suzanne W.
Numerade Educator

Problem 41

Prove that the function of the form
$$
\psi_l(\mathbf{n})=\varepsilon_{i k \ldots n} n_i n_k \ldots n_n,
$$where $\mathbf{n}=\mathbf{r} / r$ and $\varepsilon_{i k \ldots n}$ is a completely symmetric tensor ${ }^{[29]}$ of the $l$ th rank with a trace $\varepsilon_{i i k \ldots n}=0$, is an eigenfunction of the squared angular momentum operator of a particle whose angular momentum is $l$.

Further, show that the number of independent tensor components is equal to $2 l+1$, which coincides with the number of the spherical functions, $Y_{l m}(\mathbf{n})$. This shows that the angular dependence given above is the most general such dependence for the particle states with the angular momentum, $l$.

For the special cases where $l=1$ and $l=2$, indicate explicit expressions for the corresponding tensor components $\varepsilon_{\mathbf{i}}(m)$ and tensors $\varepsilon_{i k}(m)$ that make the wavefunction (1) equal to the spherical function $Y_{l m}$.

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Problem 42

According to the previous problem, the most general angular dependence of a state with the angular momentum $l=1$ has the form $\psi_{l=1}=\varepsilon \cdot \mathbf{n}$, where $\varepsilon$ is some arbitrary complex vector. Find
a) a condition on the vector $\varepsilon$ for the wavefunction to be normalized to unity;
b) mean values of the tensor components $\overline{n_i n_k}$;
c) mean values of the angular momentum vector components $\overline{\mathbf{l}}$;
d) a condition on the vector $\varepsilon$ for being able to find such a $\tilde{z}$-axis in space that angular momentum $\tilde{z}$-projection has a definite value $\tilde{m}=0$ or $\tilde{m}= \pm 1$.

Lainey Roebuck
Lainey Roebuck
Numerade Educator

Problem 43

For the conditions of the previous problem, find the probabilities, $w(\tilde{m})$, of different values of $\tilde{m}$ of the angular-momentum projection on the $\tilde{z}$-axis directed along the unit vector $\tilde{\mathbf{k}}$. Show that for an arbitrary state with the angular momentum $l=1$, there exists a spatial direction that the probability of angular momentum projection $\tilde{m}=0$ onto it is equal to zero.

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01:31

Problem 44

According to Problem 3.41, the angular dependence of an arbitrary state with the angular momentum $l=1$ has the form $\psi_{l=1}=(\mathbf{a} \cdot \mathbf{n})$, i.e., it is completely determined by some complex vector $\mathbf{a}$. Therefore in the case of states with the angular momentum $l=1$, we can use a representation (let us call it the vector representation) in which the wavefunction coincides with components of the vector a, i.e., $\psi(k) \equiv a_k,(k=1,2,3)$.
Determine an explicit form of the angular momentum component operators in the vector representation.

Suzanne W.
Suzanne W.
Numerade Educator
02:19

Problem 45

For a system of two particles which have the same angular momenta $l_1=l_2=1$, indicate:
a) the most general angular dependence of the wavefunction;
b) the most general angular dependence of the wavefunction $\psi_L$ that describes the system states with a given value $L(L=0,1,2)$ of the total angular momentum;
c) the angular dependence of the wavefunctions $\psi_{L M}$ for the system states with a given value of the total angular momentum $L$ and $z$-projection $M$.
Use the results of Problem 3.41.

Dominador Tan
Dominador Tan
Numerade Educator
02:19

Problem 46

For a system of two particles, one having the angular momentum, $l_1=1$, find the angular dependence of the wavefunction, $\psi_{J J, \Lambda}$, describing system states that correspond to definite values of the total angular momentum $J=0$ and 1 , the $z$-projection $J_z$, and angular momentum projection $\Lambda$ onto the direction of the second particle's radius-vector, specifically considering $\Lambda=0$. What are the parities of these states? What are the possible angular momenta, $l_2$, of the second particle in such states? Generalize the result to the case of arbitrary values of $l_1, J, J_z$ (but $\Lambda=0$ ).

Dominador Tan
Dominador Tan
Numerade Educator
06:46

Problem 47

For a system consisting of three particles, prove that any state with total angular momentum $L=0$ (in the center-of-mass system) has a definite, positive parity.

Keshav Singh
Keshav Singh
Numerade Educator