Question

There is a set of eigenstates |φ n) for the Hermitian operator A with non-degenerate eigenvalues an and a general state IV) ŽnCn pn〉 i. Write down the equation relating the states Iøn), the operator A and the eigenvalues an in Dirac notation 11. Use Dirac notation to explain the requirement for an operator to be Hermit ian What does it imply about the eigenvalues? 111. Explain the relation between the eigenvalues of an operator and the measured quant ities for the opera tor. iv. Explain how to determine cn using Dirac notation. How do the values of cn relate to the measurements for this operator? v. For l) to be properly normalised write down the condition required for the values of cn. vi. Express the orthonormality of the states n) and |øm) using Dirac notation vii. Evaluate the expectation value of A for

Answer 1

i)

$\hat A |\phi_n>=a_n|\phi_n>$

ii) For any general operator A acting on the wave functions

$<\phi_m|\hat A |\phi_n>=<\phi_n|\hat A^\dagger|\phi_m>^*$ ....(1)

where star denotes complex conjugate

For A to be hermitian

$\hat A =\hat A^\dagger$

and

$<\phi_n|\hat A^\dagger=a_n^*<\phi_n|$

Putting n = m and using the eigenvalue equations for A and hermitian conjugate of A gives

$<\phi_n|\hat A |\phi_n>=a_n<\phi_n|\phi_n>=a_n$ ...(2)

and

$<\phi_n|\hat A^\dagger |\phi_n>=a_n^*<\phi_n|\phi_n>=a_n^*$ ...(3)

Since $\hat A =\hat A^\dagger$ equations (2) and (3) gives

$a_n=a_n^*$

which means the eigenvalues are real

iii) Eigenvalues of the operator are the observables corresponding to the operator if the wavefunction collapses to the measured eigenfunction. The observable corresponding to A is its eigenvalue.

$<\phi_n|\hat A |\phi_n>=a_n<\phi_n|\phi_n>=a_n$

iv) Since the eigenfunctions$|\phi_n>$ are orthonormal

$<\phi_n|\psi>=\sum_n c_m<\phi_n|\phi_m>=\sum_m c_m\delta_{nm}=c_n$

where we have used the orhonormal property of eigenfunctions

$<\phi_n|\phi_m>=\delta_{nm}$

and $\delta_{nm}$ is the Kronecker delta which means it is 1 when n = m and zero otherwise

Therefore

$c_n=<\phi_n|\psi>$

v) For $\psi$ to be normalised

the condition is

$<\psi|\psi>=1$

And

$<\psi|\psi>=\sum_n \sum_m c_n c_m^* <\phi_n|\phi_m>=\sum_n \sum_m c_n c_m^*\delta_{nm}=\sum_n |c_n |^2$

Therefore

$\sum_n |c_n |^2=1$

vi) The orthonormality condition is

$<\phi_n|\phi_m>=\delta_{nm}$

vii) Expectation value of A is

$<\psi|A|\psi>=\sum_m \sum_n c_m^* c_n<\phi_m |A|\phi_n>=\sum_m \sum_n c_m^* c_na_n <\phi_m |\phi_n>$

Therefore

$<A>=\sum_m \sum_n c_m^* c_na_n \delta_{mn}=\sum_n |c_n|^2 a_n$