Q.No:1 GATE-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.
Check Answer
Option A
Q.No:2 GATE-2016
Consider an elastic scattering of particles in \(l=0\) states. If the corresponding phase shift \(\delta_0\) is \(90^{\circ}\) 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 __________.
Check Answer
Ans 2
Q.No:3 GATE-2016
Protons and \(\alpha\)-particles of equal initial momenta are scattered off a gold foil in a Rutherford scattering experiment. The scattering cross sections for proton on gold and \(\alpha\)-particle on gold are \(\sigma_p\) and \(\sigma_{\alpha}\) respectively. The ratio \(\sigma_{\alpha}/\sigma_p\) is _________.
Check Answer
Ans 4
Q.No:4 GATE-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
({\it 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)
\(q^2\)
Check Answer
Option A
Q.No:5 GATE-2021
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) \( E_1 \geq E_0 \), \( E_2 \geq E_0 \), \( E_1 \geq E_2 \)
Check Answer
Option D
Q.No:6 GATE-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 the 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 \(\phi\)
Check Answer
Option d
Q.No: 1 CSIR Dec-2014
The ground state energy of the attractive delta functional potential
\[
V(x)=-b\delta(x),
\]
where \(b>0\), calculated with the variational trial function
\[
\psi(x)=
\left\{
\begin{array}{ll}
A\cos{\frac{\pi x}{2a}}, & \text{for }-a<x<a, \\
0, & \text{otherwise},
\end{array}
\right.
\]
is
(1)
\(-\frac{mb^2}{\pi^2 \hbar^2}\)
(2)
\(-\frac{2mb^2}{\pi^2 \hbar^2}\)
(3)
\(-\frac{mb^2}{2\pi^2 \hbar^2}\)
(4)
\(-\frac{mb^2}{4\pi^2 \hbar^2}\)
Check Answer
Option 2
Q.No: 2 CSIR Dec-2014
Consider a particle of mass \(m\) in the potential \(V(x)=a|x|\), \(a>0\). The energy eigen-values \(E_n\) (\(n=0, 1, 2, \cdots\)), in the WKB approximation, are
(1)
\(\left[\frac{3a\hbar \pi}{4\sqrt{2m}} \left(n+\frac{1}{2}\right)\right]^{1/3}\)
(2)
\(\left[\frac{3a\hbar \pi}{4\sqrt{2m}} \left(n+\frac{1}{2}\right)\right]^{2/3}\)
(3)
\(\frac{3a\hbar \pi}{4\sqrt{2m}} \left(n+\frac{1}{2}\right)\)
(4)
\(\left[\frac{3a\hbar \pi}{4\sqrt{2m}} \left(n+\frac{1}{2}\right)\right]^{4/3}\)
Check Answer
Option 2
Q.No: 3 CSIR June-2015
The differential cross-section for
scattering by a target is given by \(\frac{d \sigma}{d \Omega}{(\theta, \varphi)} = a^{2} + b^{2} \cos^{2} \theta\).
If \(N\) is the flux of the incoming particles, the number of particles scattered per unit time is.
(1)
\(\frac{4 \pi}{3} N(a^{2} + b^{2})\)
(2)
\(4 \pi N\bigg(a^2 + \frac{1}{6}b^{2}\bigg)\)
(3)
\(4 \pi N\bigg(\frac{1}{2}a^2 + \frac{1}{3}b^{2}\bigg)\)
(4)
\(4 \pi N\bigg(a^2 + \frac{1}{3}b^{2}\bigg)\)
Check Answer
Option 4
Q.No: 4 CSIR June-2015
A particle of energy E scatters off a repulsive spherical potential
where \(V_{0}\), and \(a\) are positive constants. In the low energy limit, the total scattering cross section is \(\sigma=4 \pi a^{2}\left(\frac{1}{k a} \tanh k a-1\right)^{2}\), where \(k^{2}=\frac{2 m}{\hbar^{2}}\left(V_{0}-E\right)>0\). In the limit \(V_{0} \rightarrow \infty\) the ratio of \(\sigma\) to the classical scattering cross-section off a sphere of radius \(a\) is
(1)
\(4\)
(2)
\(3\)
(3)
\(1\)
(4)
\(\frac{1}{2}\)
Check Answer
Option 1
Q.No: 5 CSIR Dec-2015
In the scattering of some elementary particles, the scattering cross-section \(\sigma\) is found to depend on the total energy \(E\) and the fundamental constants \(h\) (Planck's constant) and \(c\) (the speed of light in vacuum). Using dimensional analysis, the dependence of \(\sigma\) on these quantities is given by
(1)
\(\sqrt{\frac{hc}{E}}\)
(2)
\(\frac{hc}{E^{3/2}}\)
(3)
\(\left(\frac{hc}{E}\right)^2\)
(4)
\(\frac{hc}{E}\)
Check Answer
Option 3
Q.No: 6 CSIR Dec-2015
The ground state energy of a particle in the potential \(V(x)=g|x|\), estimated using the trial wavefunction
\[
\psi(x)=
\left\{
\begin{array}{ll}
\sqrt{\frac{c}{a^5}}(a^2-x^2), & x< |a| \\
0, & x\geq |a|
\end{array}
\right.
\]
(where \(g\) and \(c\) are constants) is
(1)
\(\frac{15}{16}\left(\frac{\hbar^2 g^2}{m}\right)^{1/3}\)
(2)
\(\frac{5}{6}\left(\frac{\hbar^2 g^2}{m}\right)^{1/3}\)
(3)
\(\frac{3}{4}\left(\frac{\hbar^2 g^2}{m}\right)^{1/3}\)
(4)
\(\frac{7}{8}\left(\frac{\hbar^2 g^2}{m}\right)^{1/3}\)
Check Answer
Option 1
Q.No: 7 CSIR June-2016
The energy levels for a particle of mass \(m\) in the potential \(V(x)=\alpha |x|\), determined in the WKB approximation
\[
\sqrt{2m} \int_a^b \sqrt{E-V(x)} dx=\left(n+\frac{1}{2}\right)\hbar \pi,
\]
(where \(a, b\) are the turning points and \(n=0, 1, 2 \cdots\)), are
(1)
\(E_n=\left[\frac{\hbar \pi \alpha}{4\sqrt{m}}\left(n+\frac{1}{2}\right)\right]^{2/3}\)
(2)
\(E_n=\left[\frac{3\hbar \pi \alpha}{4\sqrt{2m}}\left(n+\frac{1}{2}\right)\right]^{2/3}\)
(3)
\(E_n=\left[\frac{3\hbar \pi \alpha}{4\sqrt{m}}\left(n+\frac{1}{2}\right)\right]^{2/3}\)
(4)
\(E_n=\left[\frac{\hbar \pi \alpha}{4\sqrt{2m}}\left(n+\frac{1}{2}\right)\right]^{2/3}\)
Check Answer
Option 2
Q.No: 8 CSIR June-2016
The ground state energy of a particle of mass \(m\) in the potential \(V(x)=\frac{\hbar^2 \beta}{6m} x^4\), estimated using the normalized trial wavefunction \(\psi(x)=\left(\frac{\alpha}{\pi}\right)^{1/4} e^{-\alpha x^2/2}\), is [Use \(\sqrt{\frac{\alpha}{\pi}}\int_{-\infty}^{\infty} dx x^2 e^{-\alpha x^2}=\frac{1}{2\alpha}\) and \(\sqrt{\frac{\alpha}{\pi}}\int_{-\infty}^{\infty} dx x^4 e^{-\alpha x^2}=\frac{3}{4\alpha^2}\)].
(1)
\(\frac{3}{2m}\hbar^2 \beta^{1/3}\)
(2)
\(\frac{8}{3m}\hbar^2 \beta^{1/3}\)
(3)
\(\frac{2}{3m}\hbar^2 \beta^{1/3}\)
(4)
\(\frac{3}{8m}\hbar^2 \beta^{1/3}\)
Check Answer
Option 4
Q.No: 9 CSIR Dec-2016
A particle is scattered by a central potential \(V(r)=V_0 re^{-\mu r}\), where \(V_0\) and \(\mu\) are positive constants. If the momentum transfer \(\vec{q}\) is such that \(q=|\vec{q}|\gg \mu\), the scattering cross-section in the Born approximation, as \(q\to \infty\), depends on \(q\) as [You may use \(\int x^n e^{ax} dx=\frac{d^n}{da^n} \int e^{ax} dx\)]
(1)
\(q^{-8}\)
(2)
\(q^{-2}\)
(3)
\(q^{2}\)
(4)
\(q^{6}\)
Check Answer
Option 1
Q.No: 10 CSIR June-2017
Consider the potential
\[
V(\vec{r})=\sum_i V_0 a^3 \delta^3(\vec{r}-\vec{r}_i)
\]
where \(\vec{r}_i\) are the position vectors of the vertices of a cube of length \(a\) centered at the origin and \(V_0\) is a constant. If \(V_0 a^2\ll \frac{\hbar^2}{m}\), the total scattering cross-section, in the low-energy limit, is
(1)
\(16a^2\left(\frac{mV_0 a^2}{\hbar^2}\right)\)
(2)
\(\frac{16a^2}{\pi^2}\left(\frac{mV_0 a^2}{\hbar^2}\right)^2\)
(3)
\(\frac{64a^2}{\pi}\left(\frac{mV_0 a^2}{\hbar^2}\right)^2\)
(4)
\(\frac{64a^2}{\pi^2}\left(\frac{mV_0 a^2}{\hbar^2}\right)\)
Check Answer
Option 3
Q.No: 11 CSIR June-2017
Using the trial function
\[
\psi(x)=
\left\{
\begin{array}{cc}
A(a^2-x^2), & -a< x< a \\
0 & \text{otherwise}
\end{array}
\right.
\]
the ground state energy of a one-dimensional harmonic oscillator is
(1)
\(\hbar \omega\)
(2)
\(\sqrt{\frac{5}{14}}\hbar \omega\)
(3)
\(\frac{1}{2}\hbar \omega\)
(4)
\(\sqrt{\frac{5}{7}}\hbar \omega\)
Check Answer
Option 2
Q.No: 12 CSIR Dec-2017
A phase shift of \(30^{\circ}\) is observed when a beam of particles of energy \(0.1 MeV\) is scattered by a target. When the beam energy is changed, the observed phase shift is \(60^{\circ}\). Assuming that only \(s\)-wave scattering is relevant and that the cross-section does not change with energy, the beam energy is
(1)
\(0.4 MeV\)
(2)
\(0.3 MeV\)
(3)
\(0.2 MeV\)
(4)
\(0.15 MeV\)
Check Answer
Option 2
Q.No: 13 CSIR Dec-2017
The energy eigenvalues \(E_n\) of a quantum system in the potential \(V=cx^6\) (where \(c>0\) is a constant), for large values of the quantum number \(n\), varies as
(1)
\(n^{4/3}\)
(2)
\(n^{3/2}\)
(3)
\(n^{5/4}\)
(4)
\(n^{6/5}\)
Check Answer
Option 2
Q.No:14 CSIR June-2018
The \(n\)-th energy eigenvalue \(E_n\) of a one-dimensional Hamiltonian \(H=\frac{p^2}{2m}+\lambda x^4\) (where \(\lambda > 0\) is a constant) in the WKB approximation, is proportional to
(1)
\((n+\frac{1}{2})^{4/3} \lambda^{1/3}\)
(2)
\((n+\frac{1}{2})^{4/3} \lambda^{2/3}\)
(3)
\((n+\frac{1}{2})^{5/3} \lambda^{1/3}\)
(4)
\((n+\frac{1}{2})^{5/3} \lambda^{2/3}\)
Check Answer
Option 1
Q.No:15 CSIR June-2018
The differential scattering cross-section \(\frac{d\sigma}{d\Omega}\) for the central potential \(V(r)=\frac{\beta}{r}e^{-\mu r}\), where \(\beta\) and \(\mu\) are positive constants, is calculated in the first Born approximation. Its dependence on the scattering angle \(\theta\) is proportional to (\(A\) is a constant below.)
(1)
\((A^2+\sin^2{\frac{\theta}{2}})\)
(2)
\((A^2+\sin^2{\frac{\theta}{2}})^{-1}\)
(3)
\((A^2+\sin^2{\frac{\theta}{2}})^{-2}\)
(4)
\((A^2+\sin^2{\frac{\theta}{2}})^{2}\)
Check Answer
Option 3
Q.No:16 CSIR June-2019
The Hamiltonian of a quantum particle of mass \(m\) is \(H=\frac{p^2}{2m}+\alpha |x|^r\), where \(\alpha\) and \(r\) are positive constants. The energy \(E_n\) of the \(n^{\text{th}}\) level, for large \(n\), depends on \(n\) as
(1)
\(n^{2r}\)
(2)
\(n^{r+2}\)
(3)
\(n^{1/(r+2)}\)
(4)
\(n^{2r/(r+2)}\)
Check Answer
Option 4
Q.No:17 CSIR June-2019
In the partial wave expansion, the differential scattering cross-section is given by
\[
\frac{d\sigma}{d(\cos{\theta})}=\left|\sum_{l} (2l+1)e^{i\delta_l} \sin{\delta_l} P_l(\cos{\theta})\right|^2
\]
where \(\theta\) is the scattering angle. For a certain neutron-nucleus scattering, it is found that the two lowest phase shifts \(\delta_0\) and \(\delta_1\) corresponding to \(s\)-wave and \(p\)-wave, respectively, satisfy \(\delta_1 \approx \delta_0/2\). Assuming that the other phase shifts are negligibly small, the differential cross-section reaches its minimum for \(\cos{\theta}\) equal to
(1)
\(0\)
(2)
\(\pm 1\)
(3)
\(-\frac{2}{3}\cos^2{\delta_1}\)
(4)
\(\frac{1}{3}\cos^2{\delta_1}\)
Check Answer
Option 3
Q.No: 18 CSIR June-2019
The elastic scattering of a charged particle of mass \(m\) off an atom can be approximated by the potential \(V(r)=\frac{\alpha}{r} e^{-r/R}\), where \(\alpha\) and \(R\) are positive constants. If the wave number of the incoming particle is \(k\) and the scattering angle is \(2\theta\), the differential cross-section in the Born approximation is
(1)
\(\frac{m^2 \alpha^2 R^4}{4\hbar^4(1+k^2 R^2 \sin^2{\theta})}\)
(2)
\(\frac{m^2 \alpha^2 R^4}{\hbar^4(2k^2 R^2+\sin^2{\theta})^2}\)
(3)
\(\frac{2m^2 \alpha^2 R^4}{\hbar^4(2k^2 R^2+\sin^2{2\theta})}\)
(4)
\(\frac{4m^2 \alpha^2 R^4}{4\hbar^4(1+4k^2 R^2 \sin^2{\theta})^2}\)
Check Answer
Option 4
Q.No: 19 CSIR June-2020
A particle with incoming wave vector \(\vec{k}\), after being scattered by the potential \(V(r)=\frac{c}{r^2}\), goes out with wave vector \(\vec{k'}\). The differential scattering cross-section, calculated in the first Born approximation, depends on \(q=|\vec{k}-\vec{k'}|\), as
(a)
\(1/q^2\)
(b)
\(1/q^4\)
(c)
\(1/q\)
(d)
\(1/q^{3/2}\)
Check Answer
Option a
Q.No: 20 CSIR Feb-2022
The Hamiltonian of a particle of mass \(m\) in one-dimension is \(H=\frac{1}{2m}p^2+\lambda |x|^3\), where \(\lambda>0\) is a constant. If \(E_1\) and \(E_2\) respectively, denote the ground state energies of the particle
for \(\lambda=1\) and \(\lambda=2\) (in appropriate units) the ratio \(E_1/E_2\) is best approximated by
(1)
\(1.260\)
(2)
\(1.414\)
(3)
\(1.516\)
(4)
\(1.320\)
Check Answer
Option 4
Q.No: 21 CSIR Feb-2022
In an elastic scattering process at an energy \(E\) , the phase shifts satisfy \(\delta_0 \approx 30^0 , \delta_1\approx 10^0\) ,
while the other phase shifts are zero. The polar angle at which the differential cross section peaks
is closest to
(1)
\(20^0\)
(2)
\(10^0\)
(3)
\(0^0\)
(4)
\(30^0\)
Check Answer
Option 3
Q.No: 22 CSIR June-2015
A particle of energy E scatters off a repulsive spherical potential
where \(V_{0}\), and \(a\) are positive constants. In the low energy limit, the total scattering cross section is \(\sigma=4 \pi a^{2}\left(\frac{1}{k a} \tanh k a-1\right)^{2}\), where \(k^{2}=\frac{2 m}{\hbar^{2}}\left(V_{0}-E\right)>0\). In the limit \(V_{0} \rightarrow \infty\) the ratio of \(\sigma\) to the classical scattering cross-section off a sphere of radius \(a\) is
(1)
4
(2)
3
(3)
1
(4)
1/2
Check Answer
Option 1
Q.No: 23 CSIR Dec-2023
An incident plane wave with wavenumber \( k \) is scattered by a spherically symmetric soft potential. The scattering occurs only in S- and P- waves. The approximate scattering amplitude at angles \( \theta = \frac{\pi}{3} \) and \( \theta = \frac{\pi}{2} \) are
\(
f \left( \theta = \frac{\pi}{3} \right) \simeq \frac{1}{2k} \left( \frac{5}{2} + 3i \right) \quad \text{and} \quad f \left( \theta = \frac{\pi}{2} \right) \simeq \frac{1}{2k} \left( 1 + \frac{3i}{2} \right).
\)
Then the total scattering cross-section is closest to
1) \( \frac{37\pi}{4k^2} \)
2) \( \frac{10\pi}{k^2} \)
3) \( \frac{35\pi}{4k^2} \)
4) \( \frac{9\pi}{k^2} \)
Check Answer
Option 1
Q.No: 24 CSIR June-2023
The phase shifts of the partial waves in an elastic scattering at energy \(E\) are \(\delta_0=12^0\), \(\delta_1=4^0\) and \(\delta_{l \ge 2} \simeq 0^0\). The best qualitative depiction of \(\theta\)-dependence of the differential scattering cross-section \(\frac{d\sigma}{d\cos\theta}\) is
Check Answer
Option b
Q.No: 26 CSIR June-2024
The Hamiltonian of a particle of mass \( m \) is given by
\[
H = \frac{p^2}{2m} + V(x),
\]
with
\[
V(x) =
\begin{cases}
-\alpha x & \text{for } x \leq 0 \\
\beta x & \text{for } x > 0
\end{cases}
\]
where \( \alpha, \beta \) are positive constants. The \( n^{th} \) energy eigenvalue \( E_n \) obtained using WKB approximation is
\[
E_n^{3/2} = \frac{3}{2} \left(\frac{\hbar^2}{2m}\right)^{1/2} \pi \left(n - \frac{1}{2}\right) f(\alpha, \beta) \quad (n = 1, 2, \ldots).
\]
The function \( f(\alpha, \beta) \) is
1) \( \frac{\sqrt{\alpha^2 \beta^2}}{\sqrt{2(\alpha^2 + \beta^2)}} \)
2) \( \frac{\alpha \beta}{\alpha + \beta} \)
3) \( \frac{\alpha + \beta}{4} \)
4) \( \frac{1}{2} \sqrt{\frac{\alpha^2 + \beta^2}{2}} \)
Check Answer
Option 2
Q.No: 27 CSIR June-2024
In a scattering experiment, a beam of \( e^- \) with an energy of 420 MeV scatters off an atomic nucleus. If the first minimum of the differential cross section is observed at a scattering angle of 45°, the radius of the nucleus (in fermi) is closest to
1) 0.4
2) 8.0
3) 2.5
4) 0.8
Check Answer
Option 3
Q.No: 28 CSIR June-2025
When a neutron of 1 keV kinetic energy impinges on a \(^{12}\mathrm{C}\) target, the total scattering cross section is 1000 barns. The approximate value of the phase shift \(\delta_{0}\) is:
1) \(18^\circ\)
2) \(108^\circ\)
3) \(90^\circ\)
4) \(36^\circ\)
Check Answer
Option 4
Q.No: 1 JEST-2017
Consider a particle confined by a potential \(V(x)=k|x|\), where \(k\) is a positive constant. The spectrum \(E_n\) of the system, within the WKB approximation, is proportional to
(A)
\((n+\frac{1}{2})^{3/2}\).
(B)
\((n+\frac{1}{2})^{2/3}\).
(C)
\((n+\frac{1}{2})^{1/2}\).
(D)
\((n+\frac{1}{2})^{4/3}\).
Check Answer
Option B
Q.No: 1 TIFR-2014
A particle \(P\) of mass \(m\) moves under the influence of a central potential, centred at the origin \(O\), of the form
\[
V(r)=-\frac{k}{3r^3}
\]
where \(k\) is a positive constant
(includegraphics)
If the particle \(P\) comes in from infinity with initial velocity \(u\) and impact parameter \(b\) (see figure), then the largest value of \(b\) for which the particle gets captured by the potential is
(a)
\(\left(\frac{3k^2}{m^2 u^4}\right)^{1/6}\)
(b)
\(\left(\frac{k}{3mu}\right)^{1/3}\)
(c)
\(\left(\frac{2k^2}{m^2 u^4}\right)^{1/6}\)
(d)
\(\left(\frac{2k}{3mu}\right)^{1/3}\)
Check Answer
Option a
Q.No: 2 TIFR-2016
In a Rutherford scattering experiment, the number \(N\) of particles scattered in a direction \(\theta\), i.e. \(dN/d\theta\), as a function of the scattering angle \(\theta\) (in the laboratory frame) varies as
(a)
\(\csc^{4}{\frac{\theta}{2}}\)
(b)
\(\csc^{2}{\frac{\theta}{2}}\cot{\frac{\theta}{2}}\)
(c)
\(\csc^{2}{\frac{\theta}{2}}\tan^{2}{\frac{\theta}{2}}\)
(d)
\(\sec^{4}{\frac{\theta}{2}}\)
Check Answer
Option b
Q.No: 3 TIFR-2023
Consider a particle of mass \(m\) in a quartic potential
\[H=\frac{p^2}{2m}+ax^4\]
If we take a variational wavefunction
\[\psi(x,\lambda)=e^{-\lambda x^2}\]
with \(\lambda >0\) and try to estimate the ground state energy, the value of \(\lambda\) should be chosen as
[You may use the integral
\[\int_{-\infty}^{+\infty} dx(A+Bx^2+Cx^4)e^{-\lambda x^2}=A\sqrt{\frac{\pi}{\lambda}}+\frac{B}{2}\sqrt{\frac{\pi}{\lambda^3}}+\frac{3C}{4}\sqrt{\frac{\pi}{\lambda^5}}\]
where \(A,B,C\) and \(\lambda>0\) are all constants.]
(a)
\((\frac{5 ma}{3\pi^2 \hbar^2})^{1/3}\)
(b)
\((\frac{15 ma}{8 \hbar^2})^{1/3}\)
(c)
\((\frac{ ma}{2\pi \hbar^2})^{1/3}\)
(d)
\((\frac{3ma}{4 \hbar^2})^{1/3}\)
Check Answer
Option d
Q.No: 4 TIFR-2024
The Hamiltonian for a Helium atom is given as \( H = H_0 + H_1 \), where
\[ H_0 = \frac{(p_1^2 + p_2^2)}{2\mu} - \frac{2e^2}{4\pi\epsilon_0 r_1} - \frac{2e^2}{4\pi\epsilon_0 r_2} \]
and
\[ H_1 = \frac{e^2}{4\pi\epsilon_0 r_{12}} \]
where \( \mu \) is the reduced mass of the electron, \( r_1 \) and \( r_2 \) are the distance of the electrons from the nucleus, and \( r_{12} \) is the distance between the two electrons. The value of the first ionization potential of the Helium atom is 24.6 eV.
What is the correction due to \( H_1 \) to the ground state energy of the Helium atom, compared to \( H_0 \)?
(a) 29.8 eV
(b) -29.8 eV
(c) 84.2 eV
(d) -2.6 eV
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