Quantum Mechanics | Part IB, 2003

What is the significance of the expectation value

Q=ψ(x)Qψ(x)dx\langle Q\rangle=\int \psi^{*}(x) Q \psi(x) d x

of an observable QQ in the normalized state ψ(x)\psi(x) ? Let QQ and PP be two observables. By considering the norm of (Q+iλP)ψ(Q+i \lambda P) \psi for real values of λ\lambda, show that

Q2P214[Q,P]2\left\langle Q^{2}\right\rangle\left\langle P^{2}\right\rangle \geqslant \frac{1}{4}|\langle[Q, P]\rangle|^{2}

The uncertainty ΔQ\Delta Q of QQ in the state ψ(x)\psi(x) is defined as

(ΔQ)2=(QQ)2.(\Delta Q)^{2}=\left\langle(Q-\langle Q\rangle)^{2}\right\rangle .

Deduce the generalized uncertainty relation,

ΔQΔP12[Q,P].\Delta Q \Delta P \geqslant \frac{1}{2}|\langle[Q, P]\rangle| .

A particle of mass mm moves in one dimension under the influence of the potential 12mω2x2\frac{1}{2} m \omega^{2} x^{2}. By considering the commutator [x,p][x, p], show that the expectation value of the Hamiltonian satisfies

H12ω.\langle H\rangle \geqslant \frac{1}{2} \hbar \omega .

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