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GATE 2024 PH – Question 13

Quantum Mechanics · one dimensional potentials: step potential, finite rectangular well, tunnelling from a potential barrier, particle in 1,2,3-dimensional box, particle in single and double delta function potentials, 1,2,3 dimensional harmonic oscillator: concept of degeneracy · 1 mark · Multiple choice

A particle in an infinite one-dimensional well has $$\Psi(x,t)=\sqrt{\frac23}e^{-iE_1t/\hbar}\psi_1(x)+\frac1{\sqrt6}e^{i\pi/6}e^{-iE_2t/\hbar}\psi_2(x)+\frac1{\sqrt6}e^{i\pi/4}e^{-iE_3t/\hbar}\psi_3(x).$$ The normalized $\psi_1,\psi_2,\psi_3$ are the ground, first excited and second excited states, with energies $E_1,E_2,E_3$. The expectation value of energy is

  1. $17E_1/6$
  2. $2E_1/3$
  3. $3E_1/2$
  4. $14E_1$

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Correct answer: (A) $17E_1/6$

Explanation

The wave function is a superposition of stationary states $\psi_1,\psi_2,\psi_3$. For orthonormal energy eigenstates the expectation value of the energy is the sum of the energies weighted by the probabilities $|c_n|^2$. The phases do not matter.

**Probabilities:**
- $|c_1|^2=\left(\sqrt{2/3}\right)^2=\dfrac23$
- $|c_2|^2=\left(\dfrac1{\sqrt6}\right)^2=\dfrac16$
- $|c_3|^2=\dfrac16$

(They add up to $\tfrac23+\tfrac16+\tfrac16=1$ ✓.)

**Energies in an infinite well:** $E_n=n^2E_1$, so $E_2=4E_1$ and $E_3=9E_1$.

$$\langle E\rangle=\frac23E_1+\frac16(4E_1)+\frac16(9E_1)=\left(\frac46+\frac46+\frac96\right)E_1=\frac{17}{6}E_1\quad(\text{option A}).$$