- #1
bjnartowt
- 284
- 3
Homework Statement
Suppose that |i> and |j> are eigenkets of some Hermitian operator. Under what condition can we conclude that |i> + |j> is also an eigenket of A? Justify your answer.
Homework Equations
It seems that all that is needed is for "A" to be a linear operator and for |i> and |j> to have the same eigenvalue. Justification:
A(u + v) = Au + Av (that's A's linearity at work)
Au = u[0]*u
Av = v[0]*v
If we have u[0] = v[0] (matching eigenvalues), then:
A(u + v) = u[0]*(u + v) = v[0]*(u + v)
...and thus:
A(u + v) = [some common scalar]*(u + v)
...meaning (u + v) is eigenstate of linear operator "A".
My question: qed?