# Meaning of bra state

1. Jan 30, 2012

### intervoxel

What exactly is a bra state. I know that it is the dual state of some ket state. Can it be interpreted as a state with negative energy?

2. Jan 30, 2012

### bhobba

No its not that ie negative energy. Given any vector space you can define linear complex valued functions on that space and they form a vector space called its dual. For the spaces used in QM it is assumed they can be put in one to one correspondence with the original vectors. When that can be done its called a Hilbert Space. In the Dirac notation the original vectors are called Kets and are denoted by |u>. The linear functions are denoted by <u| and are called Bras. Since the Bras are really linear functions defined on the Kets you can operate on a Ket with a Bra to give a complex number and that is written as <u1|u2>. It is also assumed (I am not sure if its an assumption or can be derived - I would need to look it up - its associated with something called the Rietz Representation Theorem and something in the back of my mind says the following properties pop out of it) that when written this way it has the properties of an inner product:
http://en.wikipedia.org/wiki/Inner_product_space

For the Rietz Representation Theroem check out:
http://en.wikipedia.org/wiki/Riesz_representation_theorem

Yea it looks like you can prove the inner product properties rather than assume it.

The deep reason why its introduced is because of a Theorem known as Gleasons Theorem:
http://en.wikipedia.org/wiki/Gleason's_theorem

The above is a bit technical but roughly it says the only way to define a probability on complex vector spaces is by |<U1|U2>|^2 (called the Born Rule) where it gives the probability of observing a system in state |U1> and the outcome is |U2> - |U1> and |U2> normalised.

Thanks
Bill

Last edited: Jan 30, 2012
3. Feb 1, 2012