Approximation Methods and Scattering - Quantum Mechanics - Physics Previous Year Questions

1

2010 · Physics · Quantum Mechanics · Approximation Methods and Scattering

Physics (PH) 2010

A particle of mass \( m \) is confined in an infinite potential well: \( V(x) = \begin{cases} 0 & \text{if } 0 < x < L, \\ \infty & \text{otherwise.} \end{cases} \) It is subjected to a perturbing potential \( V_p(x) = V_0 \sin\left(\frac{2\pi x}{L}\right) \) within the well. Let \( E^{(1)} \) and \( E^{(2)} \) be the corrections to the ground state energy in the first and second order in \( V_0 \), respectively. Which of the following are true?

A

\(E^{(1)} = 0; \quad E^{(2)} < 0 \)

B

\(E^{(1)} > 0; \quad E^{(2)} = 0 \)

C

\(E^{(1)} = 0; \quad E^{(2)} \) depends on the sign of \(V_0 \)

D

\(E^{(1)} < 0; \quad E^{(2)} < 0 \)

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2

2011 · Physics · Quantum Mechanics · Approximation Methods and Scattering

Physics (PH) 2011

The normalized eigenstates of a particle in a one-dimensional potential well \( V(x) = \begin{cases} 0 & \text{if } 0 \le x \le a \\ \infty & \text{otherwise} \end{cases} \) are given by \( \psi_n(x) = \sqrt{\frac{2}{a}} \sin\left(\frac{n\pi x}{a}\right) \), where \( n = 1,2,3,\cdots \). The particle is subjected to a perturbation \( V'(x) = V_0 \cos\left(\frac{\pi x}{a}\right) \) for \( 0 \le x \le \frac{a}{2} \) \( = 0 \) otherwise. The shift in the ground state energy due to the perturbation, in the first order perturbation theory, is

A

\(\frac{2V_0}{3\pi} \)

B

\(\frac{V_0}{3\pi} \)

C

\(-\frac{V_0}{3\pi} \)

D

\(-\frac{2V_0}{3\pi} \)

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3

2013 · Physics · Quantum Mechanics · Approximation Methods and Scattering

Physics (PH) 2013

A pair of eigenvalues of the perturbed Hamiltonian, using first order perturbation theory, is

A

3+2ε, 7+2ε

B

3+2ε, 2+ε

C

3, 7+2ε

D

3, 2+2ε

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4

2014 · Physics · Quantum Mechanics · Approximation Methods and Scattering

Physics (PH) 2014

A particle is confined to a one dimensional potential box with the potential
\(V(x) = 0, \quad 0 < x < a\)
\(= \infty, \quad \text{otherwise}\)
If the particle is subjected to a perturbation, within the box, W = βx, where β is a small constant, the first order correction to the ground state energy is

A

0

B

aβ/4

C

aβ/2

D

aβ

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5

2016 · Physics · Quantum Mechanics · Approximation Methods and Scattering

Physics (PH) 2016

The scattering of particles by a potential can be analyzed by Born approximation. In particular, if the scattered wave is replaced by an appropriate plane wave, the corresponding Born approximation is known as the first Born approximation. Such an approximation is valid for

A

large incident energies and weak scattering potentials.

B

large incident energies and strong scattering potentials.

C

small incident energies and weak scattering potentials.

D

small incident energies and strong scattering potentials.

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6

2016 · Physics · Quantum Mechanics · Approximation Methods and Scattering

Physics (PH) 2016

Consider an elastic scattering of particles in l = 0 states. If the corresponding phase shift \( \delta_0 \) is \( 90^0 \) and the magnitude of the incident wave vector is equal to \( \sqrt{2}\pi \) fm^{-1} then the total scattering cross section in units of fm^2 is ______.

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7

2016 · Physics · Quantum Mechanics · Approximation Methods and Scattering

Physics (PH) 2016

A hydrogen atom is in its ground state. In the presence of a uniform electric field \( \vec{E} = E_0 \hat{z} \), the leading order change in its energy is proportional to \( (E_0)^n \). The value of the exponent n is ______.

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8

2017 · Physics · Quantum Mechanics · Approximation Methods and Scattering

Physics (PH) 2017

A one dimensional simple harmonic oscillator with Hamiltonian \(H_0 = \frac{p^2}{2m} + \frac{1}{2} k x^2\) is subjected to a small perturbation, \(H_1 = \alpha x + \beta x^3 + \gamma x^4\). The first order correction to the ground state energy is dependent on

A

only \(\beta\)

B

\(\alpha\) and \(\gamma\)

C

\(\alpha\) and \(\beta\)

D

only \(\gamma\)

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9

2018 · Physics · Quantum Mechanics · Approximation Methods and Scattering

Physics (PH) 2018

The ground state energy of a particle of mass \(m\) in an infinite potential well is \(E_0\). It changes to \(E_0(1 + \alpha \times 10^{-3})\), when there is a small potential bump of height \(V_0 = \frac{\pi^2 \hbar^2}{50mL^2}\) and width \(a = L/100\), as shown in the figure. The value of \(\alpha\) is ______ (up to two decimal places).

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10

2019 · Physics · Quantum Mechanics · Approximation Methods and Scattering

Physics (PH) 2019

The Hamiltonian of a system is \( H = \begin{pmatrix} 1 & \epsilon \\ \epsilon & -1 \end{pmatrix} \) with \( \epsilon \ll 1 \). The fourth order contribution to the ground state energy of \( H \) is \( \gamma \epsilon^4 \). The value of \( \gamma \) (rounded off to three decimal places) is __________

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11

2020 · Physics · Quantum Mechanics · Approximation Methods and Scattering

Physics (PH) 2020

Consider the Hamiltonian \( \hat{H}=\hat{H}_0+\hat{H}' \) where
\( \hat{H}_0=\begin{pmatrix} E & 0 & 0 \\ 0 & E & 0 \\ 0 & 0 & E \end{pmatrix} \) and \( \hat{H}' \) is the time independent perturbation given by
\( \hat{H}'=\begin{pmatrix} 0 & k & 0 \\ k & 0 & k \\ 0 & k & 0 \end{pmatrix} \), where \( k>0 \). If the maximum energy eigenvalue of \( \hat{H} \) is 3 eV corresponding to \( E=2 \) eV, the value of \( k \) (rounded off to three decimal places) in eV is ______.

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12

2021 · Physics · Quantum Mechanics · Approximation Methods and Scattering

Physics (PH) 2021

Consider a particle in a one-dimensional infinite potential well with its walls at \( x = 0 \) and \( x = L \). The system is perturbed as shown in the figure. The first order correction to the energy eigenvalue is

A

\(\frac{V_0}{4} \)

B

\(\frac{V_0}{3} \)

C

\(\frac{V_0}{2} \)

D

\(\frac{V_0}{5} \)

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13

2021 · Physics · Quantum Mechanics · Approximation Methods and Scattering

Physics (PH) 2021

Consider the potential \( U(r) \) defined as \[ U(r) = -U_0 \frac{e^{-\alpha r}}{r} \] where \( \alpha \) and \( U_0 \) are real constants of appropriate dimensions. According to the first Born approximation, the elastic scattering amplitude calculated with \( U(r) \) for a (wave-vector) momentum transfer \( q \) and \( \alpha \to 0 \), is proportional to (\( Useful \ integral: \) \( \int_0^\infty \sin(qr) e^{-\alpha r} dr = \frac{q}{\alpha^2+q^2} \))

A

\(q^{-2} \)

B

\(q^{-1} \)

C

\(q \)

D

\(q^2 \)

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14

2022 · Physics · Quantum Mechanics · Approximation Methods and Scattering

Physics (PH) 2022

A particle of mass m in the x-y plane is confined in an infinite two-dimensional well with vertices at (0, 0), (0, L), (L, L), (L, 0). The eigenfunctions of this particle are ψ_nxny = ²/_L sin (^n_xπx/_L) sin (^n_yπy/_L). If perturbation of the form V = Cxy, where C is a real constant, is applied, then which of the following statements are correct for the first excited state?

A

The unperturbed energy is ^3π2ħ2/_2mL²

B

The unperturbed energy is ^5π2ħ2/_2mL²

C

First order energy shift due to the applied perturbation is zero

D

The shift (δ) in energy due to the applied perturbation is determined by an equation of the form |^{a-δ b}_{b a-δ}| = 0, where a and b are real, non-zero constants

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15

2024 · Physics · Quantum Mechanics · Approximation Methods and Scattering

Physics (PH) 2024

Following trial wavefunctions
\[ \phi_1 = e^{-Z'(r_1+r_2)} \]
and
\[ \phi_2 = e^{-Z'(r_1+r_2)}(1 + g|\vec{r}_1 - \vec{r}_2|) \]
are used to get a variational estimate of the ground state energy of the helium atom. \( Z' \) and \( g \) are the variational parameters, \( \vec{r}_1 \) and \( \vec{r}_2 \) are the position vectors of the electrons. Let \( E_0 \) be the exact ground state energy of the helium atom. \( E_1 \) and \( E_2 \) are the variational estimates of the ground state energy of the helium atom corresponding to \( \phi_1 \) and \( \phi_2 \), respectively. Which one of the following options is true?

A

\(E_1 \leq E_0 \), \(E_2 \leq E_0 \), \(E_1 \geq E_2 \)

B

\(E_1 \geq E_0 \), \(E_2 \leq E_0 \), \(E_1 \geq E_2 \)

C

\(E_1 \leq E_0 \), \(E_2 \geq E_0 \), \(E_1 \leq E_2 \)

D

\(E_1 \geq E_0 \), \(E_2 \geq E_0 \), \(E_1 \geq E_2 \)

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16

2024 · Physics · Quantum Mechanics · Approximation Methods and Scattering

Physics (PH) 2024

A particle of mass \(m\) in an infinite potential well of width \(a\) is subjected to a perturbation, \(V' = \frac{\hbar^2}{40ma^2}\) as shown in figure, where \(\hbar\) is Planck's constant.
The first order energy shift of the fourth energy eigenstate due to this perturbation is
\[\left(\frac{\hbar^2}{Nma^2}\right)\]
The value of \(N\) is _____ (in integer).

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17

2025 · Physics · Quantum Mechanics · Approximation Methods and Scattering

Physics (PH) 2025

A particle is scattered from a potential \(V(\vec{r}) = g\delta^3(\vec{r})\), where \(g\) is a positive constant. Using the first Born approximation, the angular \((\theta, \phi)\) dependence of differential scattering cross section \(\frac{d\sigma}{d\Omega}\) is

A

Independent of \(\theta\) but dependent on \(\phi\)

B

Dependent on \(\theta\) but independent of \(\phi\)

C

Dependent on both \(\theta\) and \(\phi\)

D

Independent of both \(\theta\) and \(\phi\)

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18

2025 · Physics · Quantum Mechanics · Approximation Methods and Scattering

Physics (PH) 2025

A two-level quantum system has energy eigenvalues \(E_1\) and \(E_2\). A perturbing potential \(H' = \lambda \Delta \sigma_x\) is introduced, where \(\Delta\) is a constant having dimensions of energy, \(\lambda\) is a small dimensionless parameter, and \(\sigma_x = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}\). The magnitudes of the first and the second order corrections to \(E_1\) due to \(H'\), respectively, are

A

0 and \(\frac{\lambda^2 \Delta^2}{|E_1 - E_2|}\)

B

\(\frac{|\lambda \Delta|}{2}\) and \(\frac{\lambda^2 \Delta^2}{|E_1 - E_2|}\)

C

\(|\lambda \Delta|\) and \(\frac{\lambda^2 \Delta^2}{|E_1 - E_2|}\)

D

0 and \(\frac{1}{2} \frac{\lambda^2 \Delta^2}{|E_1 - E_2|}\)

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