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GATE 2025 CY – Question 28

Physical Chemistry · Structure: Postulates of quantum mechanics. Operators. Commutation relations and expectations values; the role of the wave function and its absolute squares, time dependent and time independent Schrödinger equations. Born interpretation. Dirac bra-ket notation. Examples of model solvable quantum mechanical systems. · 1 mark · Multiple select

Two noninteracting particles are confined to a one-dimensional infinite box. Their wavefunctions are $\psi_1,\psi_2$ and energies $E_1,E_2$. The INCORRECT statement(s) about this system is/are

  1. The total energy is $E_1+E_2$
  2. The total wavefunction is $\psi_1+\psi_2$
  3. The total energy is $E_1E_2$
  4. The total wavefunction is $\psi_1\psi_2$

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Correct answer: (B) The total wavefunction is $\psi_1+\psi_2$; (C) The total energy is $E_1E_2$

Explanation

Two **non-interacting** particles in a box: the Hamiltonian is the sum of two independent one-particle Hamiltonians,
$$\hat H=\hat H_1+\hat H_2 .$$

**Separation of variables.** The eigenfunctions of such a sum are **products**, and the eigenvalues are **sums**:
$$\Psi=\psi_1\psi_2,\qquad E=E_1+E_2 .$$

Check each statement (the question asks for the **incorrect** ones):
- **A. The total energy is $E_1+E_2$.** Correct. ✗ (not the answer)
- **B. The total wavefunction is $\psi_1+\psi_2$.** Incorrect: a sum is not an eigenfunction of the two-particle Hamiltonian in general. ✓
- **C. The total energy is $E_1E_2$.** Incorrect: the energies add, they do not multiply. ✓
- **D. The total wavefunction is $\psi_1\psi_2$.** Correct. ✗

Answer **B and C**.