Expectation value of an observable

[Moneer81]

Homework Statement

We have an observable A, that has eigen vectors l a1 > and l a2 > , with eigenvalues a1 and a2 respectively. A second observable B has eigenvectors l b1 > and l b2 > with eigenvalues b1 and b2 respectively. The eigenstates of B can be written in terms of the eigenstates of A as:

l b1 > = 3/4 l a1 > + sqrt(7)/4 l a2 >
l b2 > = -sqrt(7)/4 l a1 > -3/4 l a2 >

If the particle is in state l b2 > , what are the expectation values <A> and <B> in terms of a1, a2, b1 and b2

Homework Equations

The Attempt at a Solution

expectiation value of, say A, is given by:

<A> = <b2 l A l b2 > since the particle is in the state b2

now how do I proceed to find <A> ? Is A a 2x2 matrix?

 

[nrqed]

Homework Statement

We have an observable A, that has eigen vectors l a1 > and l a2 > , with eigenvalues a1 and a2 respectively. A second observable B has eigenvectors l b1 > and l b2 > with eigenvalues b1 and b2 respectively. The eigenstates of B can be written in terms of the eigenstates of A as:

l b1 > = 3/4 l a1 > + sqrt(7)/4 l a2 >
l b2 > = -sqrt(7)/4 l a1 > -3/4 l a2 >

If the particle is in state l b2 > , what are the expectation values <A> and <B> in terms of a1, a2, b1 and b2

Homework Equations

The Attempt at a Solution

expectiation value of, say A, is given by:

<A> = <b2 l A l b2 > since the particle is in the state b2

now how do I proceed to find <A> ? Is A a 2x2 matrix?

Just apply A to |b2> using the fact that

A |a1> = a1 |a1>
and

A |a2> = a2 |a2>

and then using orthonormality of |a1> and |a2>