GATE 2026 CY – Question 61

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

Two atoms have spherical coordinates $(1,\pi/2,\pi/2)$ and $(1,\pi/4,3\pi/2)$, with distance in Å and angles in radians. Their separation in Å (two decimal places) is _____

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Correct answer: 1.84 to 1.86

Explanation

**Convert the spherical coordinates $(r,\theta,\phi)$ to Cartesian coordinates:**
$$x=r\sin\theta\cos\phi,\quad y=r\sin\theta\sin\phi,\quad z=r\cos\theta .$$

**Atom 1:** $(1,\ \pi/2,\ \pi/2)$:
$$x=1\cdot1\cdot0=0,\quad y=1\cdot1\cdot1=1,\quad z=0\ \Rightarrow\ (0,\ 1,\ 0).$$

**Atom 2:** $(1,\ \pi/4,\ 3\pi/2)$, with $\sin\pi/4=\cos\pi/4=\tfrac1{\sqrt2}$, $\cos3\pi/2=0$, $\sin3\pi/2=-1$:
$$x=\tfrac1{\sqrt2}\cdot0=0,\quad y=\tfrac1{\sqrt2}(-1)=-\tfrac1{\sqrt2},\quad z=\tfrac1{\sqrt2}\ \Rightarrow\ \left(0,\ -\tfrac1{\sqrt2},\ \tfrac1{\sqrt2}\right).$$

**Distance:**
$$d^2=0+\left(1+\tfrac1{\sqrt2}\right)^2+\left(\tfrac1{\sqrt2}\right)^2=1+\sqrt2+\tfrac12+\tfrac12=2+\sqrt2 .$$

$$d=\sqrt{2+\sqrt2}=\mathbf{1.85\ \text{\AA}}.$$