Central Potentials and Angular Momentum - Quantum Mechanics - Physics Previous Year Questions

1

2007 · Physics · Quantum Mechanics · Central Potentials and Angular Momentum

Physics (PH) 2007

An atomic state of hydrogen is represented by the following wavefunction: \( \psi(r, \theta, \phi) = \frac{1}{\sqrt{2}} \left( \frac{1}{a_0} \right)^{3/2} \left( 1 - \frac{r}{2a_0} \right) e^{-r/2a_0} \cos\theta \) where \( a_0 \) is a constant. The quantum numbers of the state are

A

\(\ell = 0, m = 0, n = 1 \)

B

\(\ell = 1, m = 1, n = 2 \)

C

\(\ell = 1, m = 0, n = 2 \)

D

\(\ell = 2, m = 0, n = 3 \)

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2

2007 · Physics · Quantum Mechanics · Central Potentials and Angular Momentum

Physics (PH) 2007

An atomic state of hydrogen is represented by the following wavefunction: \( \psi(r, \theta, \varphi) = \frac{1}{\sqrt{2}} \left( \frac{1}{a_0} \right)^{3/2} \left( 1 - \frac{r}{2a_0} \right) e^{-r/2a_0} \cos\theta \), where \( a_0 \) is a constant. The quantum numbers of the state are

A

\(\ell = 0, m = 0, n = 1 \)

B

\(\ell = 1, m = 1, n = 2 \)

C

\(\ell = 1, m = 0, n = 2 \)

D

\(\ell = 2, m = 0, n = 3 \)

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3

2007 · Physics · Quantum Mechanics · Central Potentials and Angular Momentum

Physics (PH) 2007

Three operators \( X \), \( Y \) and \( Z \) satisfy the commutation relations \( [X, Y] = i\hbar Z \), \( [Y, Z] = i\hbar X \) and \( [Z, X] = i\hbar Y \). The set of all possible eigenvalues of the operator \( Z \), in units of \( \hbar \), is

A

\(\{0, \pm 1, \pm 2, \pm 3, ...\} \)

B

\(\left\{ \frac{1}{2}, 1, \frac{3}{2}, 2, \frac{5}{2}, ... \right\} \)

C

\(\left\{ 0, \pm \frac{1}{2}, \pm 1, \pm \frac{3}{2}, \pm 2, \pm \frac{5}{2}, ... \right\} \)

D

\(\left\{ -\frac{1}{2}, +\frac{1}{2} \right\} \)

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4

2008 · Physics · Quantum Mechanics · Central Potentials and Angular Momentum

Physics (PH) 2008

Let $|\psi_0\rangle$ denote the ground state of the hydrogen atom. Choose the correct statement from those given below:

A

$[L_x, L_z] |\psi_0\rangle = 0$

B

$J^2 |\psi_0\rangle = 0$

C

$\vec{L} \cdot \vec{S} |\psi_0\rangle \neq 0$

D

$[S_x, S_y] |\psi_0\rangle = 0$

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5

2008 · Physics · Quantum Mechanics · Central Potentials and Angular Momentum

Physics (PH) 2008

The radial wave function of the electrons in the state of n=1 and l=0 in a hydrogen atom is $R_{10} = \frac{2}{a_0^{3/2}} \exp\left(-\frac{r}{a_0}\right)$, $a_0$ is the Bohr radius. The most probable value of r for an electron is

A

$a_0$

B

$2a_0$

C

$4a_0$

D

$8a_0$

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6

2010 · Physics · Quantum Mechanics · Central Potentials and Angular Momentum

Physics (PH) 2010

For a spin-s particle, in the eigen basis of \(\vec{S}^2\), \(S_z\), the expectation value \(\langle s m|S_x^2|s m\rangle\) is

A

\(\hbar^2 \frac{s(s+1)-m^2}{2}\)

B

\(\hbar^2 [s(s+1)-2m^2]\)

C

\(\hbar^2 [s(s+1)-m^2]\)

D

\(\hbar^2 m^2\)

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7

2011 · Physics · Quantum Mechanics · Central Potentials and Angular Momentum

Physics (PH) 2011

The normalized ground state wavefunction of a hydrogen atom is given by \( \psi(r) = \frac{1}{\sqrt{4\pi}} \frac{2}{a^{3/2}} e^{-r/a} \), where a is the Bohr radius and r is the distance of the electron from the nucleus, located at the origin. The expectation value \( \left\langle \frac{1}{r^2} \right\rangle \) is

A

\(\frac{8\pi}{a^2} \)

B

\(\frac{4\pi}{a^2} \)

C

\(\frac{4}{a^2} \)

D

\(\frac{2}{a^2} \)

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8

2011 · Physics · Quantum Mechanics · Central Potentials and Angular Momentum

Physics (PH) 2011

The normalized ground state wavefunction of a hydrogen atom is given by \(\psi(r) = \frac{1}{\sqrt{4\pi}} \frac{2}{a^{3/2}} e^{-r/a}\), where \(a\) is the Bohr radius and \(r\) is the distance of the electron from the nucleus, located at the origin. The expectation value \(\langle \frac{1}{r^2} \rangle\) is

A

\(\frac{8\pi}{a^2}\)

B

\(\frac{4\pi}{a^2}\)

C

\(\frac{4}{a^2}\)

D

\(\frac{2}{a^2}\)

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9

2012 · Physics · Quantum Mechanics · Central Potentials and Angular Momentum

Physics (PH) 2012

The ground state wavefunction for the hydrogen atom is given by \(\psi_{100} = \frac{1}{\sqrt{4\pi}} \left(\frac{1}{a_0}\right)^{3/2} e^{-r/a_0}\), where \(a_0\) is the Bohr radius.
The plot of the radial probability density, \(P(r)\) for the hydrogen atom in the ground state is

A

Graph A

B

Graph B

C

Graph C

D

Graph D

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10

2013 · Physics · Quantum Mechanics · Central Potentials and Angular Momentum

Physics (PH) 2013

Which one of the following commutation relations is NOT CORRECT ? Here, symbols have their usual meanings.

A

[L^2, L_z] = 0

B

[L_x, L_y] = i\hbar L_z

C

[L_x, L_y] = \hbar L_x

D

[L_x, L_z] = \hbar L_y

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11

2014 · Physics · Quantum Mechanics · Central Potentials and Angular Momentum

Physics (PH) 2014

If \(\vec{L}\) is the orbital angular momentum and \(\vec{S}\) is the spin angular momentum, then \(\vec{L} \cdot \vec{S}\) does NOT commute with

A

\(S_x\)

B

\(L^2\)

C

\(S^2\)

D

\((\vec{L} + \vec{S})^2\)

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12

2014 · Physics · Quantum Mechanics · Central Potentials and Angular Momentum

Physics (PH) 2014

An electron in the ground state of the hydrogen atom has the wave function \[\Psi(\vec{r}) = \frac{1}{\sqrt{\pi a_0^3}} e^{-(r/a_0)}\] where a_0 is constant. The expectation value of the operator \hat{Q} = z^2 - r^2, where z = r cos θ is (Hint: \int_0^\infty r^n e^{-ar} dr = \frac{n!}{a^{n+1}} (n-1)! / a^{n+1})

A

-a_0^2/2

B

-a_0^2

C

-3a_0^2/2

D

-2a_0^2

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13

2014 · Physics · Quantum Mechanics · Central Potentials and Angular Momentum

Physics (PH) 2014

A particle of mass m is subjected to a potential, \[V(x, y) = \frac{1}{2} m \omega^2 (x^2 + y^2), -\infty \le x \le \infty, -\infty \le y \le \infty\] The state with energy 4ℏω is g-fold degenerate. The value of g is ________.

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14

2014 · Physics · Quantum Mechanics · Central Potentials and Angular Momentum

Physics (PH) 2014

A hydrogen atom is in the state
\[ \Psi = \sqrt{\frac{9}{21}} \psi_{200} - \sqrt{\frac{3}{21}} \psi_{310} + \sqrt{\frac{4}{21}} \psi_{321}, \]
where \(n, l, m\) in \(\psi_{nlm}\) denote the principal, orbital and magnetic quantum numbers, respectively. If \(\vec{L}\) is the angular momentum operator, the average value of \(L^2\) is __________ \(\hbar^2\).

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15

2014 · Physics · Quantum Mechanics · Central Potentials and Angular Momentum

Physics (PH) 2014

If \(L_+\) and \(L_-\) are the angular momentum ladder operators, then, the expectation value of \((L_+L_- + L_-L_+)\), in the state |l = 1, m = 1> of an atom is __________ \(\hbar^2\).

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16

2016 · Physics · Quantum Mechanics · Central Potentials and Angular Momentum

Physics (PH) 2016

Let \( |l,m \rangle \) be the simultaneous eigenstates of \( L^2 \) and \( L_z \). Here \( \vec{L} \) is the angular momentum operator with Cartesian components \( (L_x, L_y, L_z) \), \( l \) is the angular momentum quantum number and \( m \) is the azimuthal quantum number. The value of \( \langle l,0| (L_x + iL_y) |l,-1 \rangle \) is

A

\(0 \)

B

\(\hbar \)

C

\(\sqrt{2}\hbar \)

D

\(\sqrt{3}\hbar \)

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17

2019 · Physics · Quantum Mechanics · Central Potentials and Angular Momentum

Physics (PH) 2019

For a spin \(\frac{1}{2}\) particle, let \(|\uparrow\rangle\) and \(|\downarrow\rangle\) denote its spin up and spin down states, respectively. If \(|a\rangle = \frac{1}{\sqrt{2}}(|\uparrow\uparrow\rangle + |\downarrow\downarrow\rangle)\) and \(|b\rangle = \frac{1}{\sqrt{2}}(|\uparrow\downarrow\rangle - |\downarrow\uparrow\rangle)\) are composite states of two such particles, which of the following statements is true for their total spin \(S\)?

A

\(S=1\) for \(|a\rangle\), and \(|b\rangle\) is not an eigenstate of the operator \(\hat{S}^2\)

B

\(|a\rangle\) is not an eigenstate of the operator \(\hat{S}^2\), and \(S=0\) for \(|b\rangle\)

C

\(S=0\) for \(|a\rangle\), and \(S=1\) for \(|b\rangle\)

D

\(S=1\) for \(|a\rangle\), and \(S=0\) for \(|b\rangle\)

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18

2019 · Physics · Quantum Mechanics · Central Potentials and Angular Momentum

Physics (PH) 2019

The Hamiltonian for a quantum harmonic oscillator of mass \(m\) in three dimensions is
\[H = \frac{p^2}{2m} + \frac{1}{2} m \omega^2 r^2\]
where \(\omega\) is the angular frequency. The expectation value of \(r^2\) in the first excited state of the oscillator in units of \(\frac{\hbar}{m \omega}\) (rounded off to one decimal place) is __________

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19

2020 · Physics · Quantum Mechanics · Central Potentials and Angular Momentum

Physics (PH) 2020

Particle \(A\) with angular momentum \(j = \frac{3}{2}\) decays into two particles \(B\) and \(C\) with angular momenta \(j_1\) and \(j_2\), respectively. If \(\left|\frac{3}{2}, \frac{3}{2}\right\rangle_A = \alpha|1,1\rangle_B \otimes \left|\frac{1}{2}, \frac{1}{2}\right\rangle_C\), the value of \(\alpha\) is ________.

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20

2020 · Physics · Quantum Mechanics · Central Potentials and Angular Momentum

Physics (PH) 2020

The radial wave function of a particle in a central potential is given by \(R(r) = A \frac{r}{a} \exp\left(-\frac{r}{2a}\right)\), where \(A\) is the normalization constant and \(a\) is positive constant of suitable dimensions. If \(\gamma a\) is the most probable distance of the particle from the force center, the value of \(\gamma\) is ______.

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