Question

Let a photon have a properly normalized quantum state given by the state vector |30>

(i.e. in a state that's at 30 degrees from the vertical).

Rewritting this state vector in the form a|0> + b|90>, what is a? What is b?

Answer 1

The state vector |30> is in between the vectors |0> and |90>, Now to find the vector as a linear combination of vectors in |0> and |90>, we have to find the projection of the vector |30> on these vectors.

The projection of a vector B on A is given by the dot product, In Hilbert space the dot product becomes .

So we can write, the state vector |30> as,

Here , is the projection of the vector on |0>, and is the projection of the vector on |90>

Therefore we can write,