A system of five identical, non-interacting particles with mass \( m \) and spin \( \frac{3}{2} \) is confined to a one-dimensional potential well of length \( L \). If the lowest energy of the system is
\[
E_{{min}} = \frac{N \pi^2 \hbar^2}{2m L^2},
\]
the value of \( N \) (in integer) is:
Show Hint
Solution and Explanation
1. System Description:
The system consists of five identical, non-interacting particles with mass \( m \) and spin \( \frac{3}{2} \), confined to a one-dimensional potential well of length \( L \). The energy levels of the system depend on the quantum numbers and the spin configuration.
2. Energy Levels of Non-Interacting Particles in a Potential Well:
For a particle in a one-dimensional infinite potential well, the energy levels are quantized and given by:
\[ E_n = \frac{n^2 \pi^2 \hbar^2}{2m L^2}, \]
where \( n \in \mathbb{Z}^+ \) is the quantum number. Since the particles are non-interacting, the total ground state energy is obtained by occupying the lowest available states while obeying the Pauli exclusion principle.
3. Spin and Energy Configuration:
Each particle has spin \( \frac{3}{2} \), giving rise to four distinct spin projections: \( m_s = \frac{3}{2}, \frac{1}{2}, -\frac{1}{2}, -\frac{3}{2} \). As these are fermions, the Pauli exclusion principle forbids any two particles from sharing the same quantum state (i.e., same \( n \) and spin projection).
4. Finding the Lowest Energy State:
Each spatial level \( n \) can accommodate 4 fermions (due to the 4 spin states). So:
- Level \( n = 1 \): 4 particles
- Level \( n = 2 \): 1 particle
Total energy: \[ E_{\text{min}} = 4E_1 + 1E_2 = 4\left( \frac{\pi^2 \hbar^2}{2m L^2} \right) + 1\left( \frac{4\pi^2 \hbar^2}{2m L^2} \right) = \frac{(4 + 4) \pi^2 \hbar^2}{2m L^2} \] \[ \Rightarrow N = \boxed{8} \]
Top GATE PH Physics Questions
- An infinite one-dimensional lattice extends along the x-axis. At each lattice site, there exists an ion with spin 1/2. The spin can point either in the +z or -z direction only. Let \( S_P \), \( S_F \), and \( S_A \) denote the entropies of paramagnetic, ferromagnetic, and antiferromagnetic configurations, respectively. Which of the following relations is/are true?
- Two point charges of charge \( +q \) each are placed a distance \( 2d \) apart. A grounded solid conducting sphere of radius \( a \) is placed midway between them. Assume \( a^2 \ll d^2 \). Which of the following statements is/are true?
- A particle of mass $m$ is moving in the potential
\[ V(x) = \begin{cases} V_0 + \frac{1}{2} m \omega_{0P}^2 x^2 & \quad \text{if } x > 0 \\ \infty & \quad \text{if } x \leq 0 \end{cases} \]
Figures P, Q, R, and S show different combinations of the values of $ \omega_0 $ and $ V_0 $. - Decays of mesons and baryons can be categorized as weak, strong and electromagnetic decays depending upon the interactions involved in the processes. Which of the following option is/are true?
- The temperature \( T \) dependence of magnetic susceptibility \( \chi \) (Column I) of certain magnetic materials (Column II) are given below. Which of the following options is/are correct?
Column I Column II (1) 
(P) Diamagnetic (2) 
(Q) Paramagnetic (3) 
(R) Ferromagnetic (4) 
(S) Antiferromagnetic
Top GATE PH Quantum Mechanics Questions
- A particle of mass \( m \) is in a potential \( V(x) = \frac{1}{2} m \omega^2 x^2 \) for \( x > 0 \), and \( V(x) = \infty \) for \( x \leq 0 \), where \( \omega \) is the angular frequency. The ratio of the energies corresponding to the lowest energy level to the next higher level is:
- 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:
- Which of the following option(s) is/are correct for photons?
Two projectile protons \( P_1 \) and \( P_2 \), both with spin up (along the \( +z \)-direction), are scattered from another fixed target proton \( T \) with spin up at rest in the \( xy \)-plane, as shown in the figure. They scatter one at a time. The nuclear interaction potential between both the projectiles and the target proton is \( \hat{\lambda} \vec{L} \cdot \vec{S} \), where \( \vec{L} \) is the orbital angular momentum of the system with respect to the target, \( \vec{S} \) is the spin angular momentum of the system, and \( \lambda \) is a negative constant in appropriate units. Which one of the following is correct?
- Let \( |m\rangle \) and \( |n\rangle \) denote the energy eigenstates of a one-dimensional simple harmonic oscillator. The position and momentum operators are \( \hat{X} \) and \( \hat{P} \), respectively. The matrix element \( \langle m | \hat{P}\hat{X} | n \rangle \) is non-zero when:
Top GATE PH Questions
- If \( F_1(Q, q) = Qq \) is the generating function of a canonical transformation from \((p, q)\) to \((P, Q)\), then which one of the following relations is correct?
- When an unpolarized plane electromagnetic wave in a dielectric medium with refractive index \( n_1 \) is incident on a plane interface separating it from another dielectric medium with refractive index \( n_2 \) (where \( n_2 > n_1 \)), and the angle of incidence is \(\tan^{-1} \left( \frac{n_2}{n_1} \right) \), the following statement is true:
- The wavefunction of a particle in an infinite one-dimensional potential well at time \( t \) is
\[ \Psi(x, t) = \sqrt{\frac{2}{3}} e^{-iE_1 t/\hbar}\psi_1(x) + \frac{1}{\sqrt{6}} e^{i\pi/6} e^{-iE_2 t/\hbar} \psi_2(x) + \frac{1}{\sqrt{6}} e^{i\pi/4} e^{-iE_3 t/\hbar} \psi_3(x) \]where \(\psi_1\), \(\psi_2\), and \(\psi_3\) are the normalized ground state, the normalized first excited state, and the normalized second excited state, respectively. \(E_1\), \(E_2\), and \(E_3\) are the eigen-energies corresponding to \(\psi_1\), \(\psi_2\), and \(\psi_3\), respectively. The expectation value of energy of the particle in state \(\Psi(x,t)\) is - If a thermodynamical system is adiabatically isolated and experiences a change in volume under an externally applied constant pressure, then the thermodynamical potential minimized at equilibrium is the
- The mean distance between the two atoms of an HD molecule is \( r \), where H and D denote hydrogen and deuterium, respectively. The mass of the hydrogen atom is \( m_H \). The energy difference between the two lowest lying rotational states of HD in multiples of \(\frac{h^2}{m_H r^2}\) is: