Question

(a) There are a set of eigenstates ๒n) for the Hermitian operator A with non-degenerate eigenvalues an and a state |ψ Σ¡c; Write down the equation relating the states |>n), the operator A and the eigenvalues a 1. ,n ii. Using Dirac notation explain the requirement for an operator to be Hermitian iii. Explain the relation between the eigenvalues of an operator and the measured iv. For to be properly normalised show the condition required for the values V. Express the orthonormality of the states |φ, and |Pm> What does it imply about the eigenvalues? quantities for the operator. of ci. vi. Evaluate the expectation value of A for |b)

Answer 1

(i)

$\hat{A}|\phi_n\rangle=c_n|\phi_n\rangle$

(ii)

If an operator is Hermitian, then the eigenvalues are real. Requirement

$\langle\phi_m|\hat{Q}|\phi_n\rangle=\langle\phi_n|\hat{Q}^\dagger|\phi_m\rangle$

(iii)

Values measured are called expectation values. For definite eigenstates, we know that

$\hat{Q}|\phi_n\rangle=q_n|\phi_n\rangle$

what we actually measure is

$\langle\phi_n|\hat{Q}|\phi_n\rangle=\langle\phi_n|q_n|\phi_n\rangle=q_n$

(iv)

$|\psi\rangle=\sum_i c_i|\phi_i\rangle$

$\langle\psi|=\sum_j c^*_j\langle\phi_j|$

$\langle\psi|\psi\rangle=\sum_j\sum_i c_ic^*_j\langle\phi_j|\phi_i\rangle$

$1=\sum_j\sum_i c_ic^*_j\delta_{ij}$

$1=\sum_i c_ic^*_i$

$\sum_i |c_i|^2=1$

(v)

I already used this condition, but I'll write it again

$\langle\phi_m|\phi_n\rangle=\delta_{mn}$

RHS is the Kronecker delta, which is 1 iff, m=n and 0 otherwise.

(vi) We require

$\langle A\rangle=\langle\psi|\hat{A}|\psi\rangle$

Proceed as in part (iv)

$\langle A\rangle=\sum_j\sum_i c_ic^*_j\langle\phi_j|\hat{A}|\phi_i\rangle$

$\langle A\rangle=\sum_j\sum_i a_ic_ic^*_j\langle\phi_j|\phi_i\rangle$

where I used

$\hat{A}|\phi_n\rangle=c_n|\phi_n\rangle$.

$\langle A\rangle=\sum_j\sum_i a_ic_ic^*_j\delta_{ij}$

$\langle A\rangle=\sum_i a_i|c_i|^2$