Question

Question #bbbce

Answer 1

In linear algebra, one studies the notion of linear vector spaces over different scalar fields.

So assuming that you are familiar with the notion of a linear vector space, let us consider an LVS `V(F)`.

Then for two elements `u` and `v` in `V`, we can associate a scalar (in the field `F`) such that the following properties hold,

1) Complex conjugate of `(u,v) = (v,u)`

2) `(u,u) >= 0` and the equality holds only for `u = 0`

3) For four vectors `u, v, w, z` in `V` and two arbitrary scalars `a` and `b` in `F`, `(a(u + v), b(w + z)) = bar a b [(u,w) + (u,z) + (v,w) + (v,z)]`

Where `bar a` denotes complex conjugation.

If these properties hold, we call `(u,v)` an inner product of `u` and `v` and `V` is then an inner product space.

However, there is no general recipe for defining the inner product of two vectors (or functions). We just have to make sure that the properties 1-3 are satisfied.

For an example, an inner product of two wave functions `psi` and `phi` in Quantum mechanics is defined as,

`< psi|phi> = int bar psi phi d tau` where integration is over all volume of the space and `bar psi` denotes complex conjugate of `psi`.

One can easily check if the properties 1-3 hold.