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The complete set of state vectors |n> are orthonormal
<n|n> = 1
The state |Ψ> is a linear combination of states |n>
|Ψ> = Σcn |n>
The co-efficients of the expansion of |psi> in terms of the |n>
Taking the inner product with <m| : <m|Ψ> = Σcn <n|m> = Σcn δmn
Hence cn = <n|Ψ>
The operator A is Hermitian
<ϕ|A|Ψ> = <A†|ϕ|Ψ> = [<Ψ|A†|ϕ>]*
For a Hermitian operator A = A†
thus <ϕ|A|Ψ> = [<Ψ|A|ϕ>]*
What are the commutator relations?
[A, B] = - [B, A]
[A, B+C] = [A, B] + [A, C]
[A, BC] = [A, B]C + B[A, C]
[AB, C] = A[B, C]+ [A, C]B
How does an operator A commute with itself and functions of itself?
[A, A^n] = 0
If f(n) = Σ cn A^n then [A, f(n)] = 0
Expectation value of A
<A> = <Ψ|A|Ψ> = Σ Σ cm* cn an <m|n> = Σ |cn|^2 an
For two eigenvectors of A, with different eigenvalues, <m|n> = 0
A |n> = an |n>, A |m> = am |m>
<n|A|m> = am <n|m>
<An|m> = an <n|m> = am <n|m>
From the definition of Hermicity an* <n|m> = am <n|m>
Hence for an ≠ am, <m|n> = 0
What is an observable in quantum mechanics?
When does a measurement of the expectation value of an observable correspond to an eigenvalue of the observable?
An observable is a measureable property, such as energy, position, momentum.
The expectation value corresponds to the measurement for a time-independent observable - a stationary state.
What is the probability of a measurement of A giving the result an / |n>?
Pn = |cn|^2
What is meant by a stationary state in QM?
An eigenstate of a Hamiltonian - a state with no time dependence.
It has a fixed expectation value.
The time dependence of the expectation value of A, <A>
i/ħ ∫Ψ*[H, A]ΨδT + δA/δt
The operator is time dependent, the state vectors are time-independent
Ehrenfest's Theorem
The standard deviation ΔA and variance (ΔA)^2
ΔA = <A - <A>>
(ΔA)^2 = (<A - <A>)^2 = (<A - <A>)(<A - <A>)
= <A^2 - 2<A><A> + <A>^2 > = <A^2> - <A>^2