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GATE 2024 CY – Question 63

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. · 2 marks · Numerical answer

$\phi_1$ and $\phi_2$ are normalized eigenfunctions of a Hermitian operator. $|\psi\rangle=3i|\phi_1\rangle+2|\phi_2\rangle$ and $|\chi\rangle=-2i|\phi_1\rangle+5|\phi_2\rangle$. The value of $\langle\psi|\chi\rangle+\langle\chi|\psi\rangle$ is ___. (rounded off to the nearest integer)

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Correct answer: 8 to 8

Explanation

The functions $\phi_1$ and $\phi_2$ are normalised eigenfunctions of a Hermitian operator, so they are **orthonormal**: $\langle\phi_i|\phi_j\rangle=\delta_{ij}$.

**Bra of $\psi$:** the coefficients are complex-conjugated, so
$$\langle\psi|=-3i\langle\phi_1|+2\langle\phi_2| .$$

**Inner product $\langle\psi|\chi\rangle$**, with $|\chi\rangle=-2i|\phi_1\rangle+5|\phi_2\rangle$:
$$\langle\psi|\chi\rangle=(-3i)(-2i)+(2)(5)=6i^2+10=-6+10=4 .$$

**The other order** is the complex conjugate: $\langle\chi|\psi\rangle=\langle\psi|\chi\rangle^*=4$.

$$\langle\psi|\chi\rangle+\langle\chi|\psi\rangle=4+4=\mathbf{8}.$$

Official GATE 2024 answer key: https://gate2026.iitg.ac.in/doc/download/2024/CYFinalAnswerKey.pdf