Answer to Question #87632 in Quantum Mechanics for Sridhar

Question

A linear Hermitian operator A has infinite number of eigen values ai and has a complete set of orthonormal eigen state |ai> . If the system is in a state |ψ> = |a1> +i/2 |a2> the probability of getting a value a2 for A on measurement is
Answer:0.2

Answer 1

The probability $P \left( a_2 \right)$ of getting the value $a_2$ is calculated by the standard quantum-mechanical formula

$$P \left( a_2 \right) = \frac{ \left| \left\langle a_2 | \psi \right\rangle \right|^2}{\left \langle \psi | \psi \right\rangle} \, .$$

We have $\left\langle a_2 | \psi \right\rangle = i / 2$, $\left| \left\langle a_2 | \psi \right\rangle \right|^2 = |i / 2|^2 = 1/4$, $\left\langle \psi | \psi \right\rangle = |1|^2 + |i/2|^2 = 5/4$, so that

$$P \left( a_2 \right) = \frac{1/4}{5/4} = \frac{1}{5} = 0.2 \, .$$

Answer: 0.2.