Homework Help: Use of vector spaces in quantum mechanics

1. Apr 5, 2013

spaghetti3451

1. The problem statement, all variables and given/known data

In quantum mechanics, what objects are the members of the vector space V? Give an example for the case of quantum mechanics of a member of the adjoint space V' and explain how members of V' enable us to predict the outcomes of experiments.

2. Relevant equations

3. The attempt at a solution

In quantum mechanics, the members of the vector space V are wavefunctions that completely specify the quantum state of a system.

If |ψ> is a member of a V, then its adjoint is the vector <ψ| which belongs to V'.

When a member of V' is operated on the corresponding member of V to obtain a complex scalar, the square of the modulus of the complex scalar gives the probability distribution for the observable associated with the member of V.

2. Apr 6, 2013

Ackbach

What is $V$? I'm having a feeling that there are some conditions on it, such as that the vectors are normalized. Is it a Hilbert space?

3. Apr 7, 2013

Fredrik

Staff Emeritus
I'm not familiar with the term "adjoint space", but it sounds like you mean "dual space". The problem sounds a bit strange. You either know the definition of "dual space", or you don't. If you do, the problem is trivial except for the last comment about predictions, which requires knowledge of the basics of QM. It's also not true that the dual space enables us to calculate probabilities. The inner product on V (which I assume is the system's Hilbert space) gets the job done just fine, so we don't need to mention the dual space...which is usually denoted by V* rather than V'.

It's also very far from true that the dual space (or the inner product) enables us to predict the outcomes. The best we can do is to predict the frequency of a specfic result in a long series of measurements on identically prepared systems.