# Wave function

1. Feb 26, 2014

### wasi-uz-zaman

hi, from study of quantum mechanics i infer that wavefunction is a dummy function on which you apply required operator like momentum operator, energy operator etc., and their eigenvalues gives you the value of observable? i want to ask my interpretation of wavefunction is correct? or not?.
thanks

2. Feb 27, 2014

### atyy

It is not correct. The wave function is the state of the system. The time evolution of the wave function is governed by the Schroedingers equation.

3. Feb 27, 2014

### Staff: Mentor

Atyy is correct

I posted the following in another thread that carefully explains what a state is which hopefully will help. To really grasp it you need to see the two axioms of QM as detailed by Ballentine in his text (Quantum Mechanics - A Modern Development).

1. To each observation there corresponds a Hermitian operator whose eigenvalues give the possible outcomes of the observation.
2. There exists a positive operator of unit trace P such that the expected outcome of the observation associated with the observable O is E(O) = Trace (PO) - this is the Born Rule in its most general form. By definition P is called the state of the system.

In fact the Born Rule is not entirely independent of the first axiom, as to a large extent it is implied from that via Gleason's Theorem - but that would take us too far afield - I simply mention it in passing.

Also note that the state, just like probabilities, is simply an aid in calculating expected outcomes. Its not real like say an electric field etc. In some interpretations its real - but the formalism of QM is quite clear - its simply, like probabilities, an aid in calculation.

By definition states of the form |x><x| are called pure. States that are a convex sum of pure states are called mixed ie are of the form ∑ pi |xi><xi| where the pi a positive and sum to one. It can be shown all states are either pure or mixed. Applying the Born rule to mixed states shows that if you have an observation whose eigenvectors are the |xi><xi| then outcome |xi><xi| will occur with probability pi. Physically one can interpret this as a system in state |xi><xi| randomly presented for observation with probability pi. In such a case no collapse occurs and an observation reveals whats there prior to observation - many issues with QM are removed. Such states are called proper mixed states.

Pure states, being defined by a single element of a vector space, can be associated with those elements and that's what's usually done. Of course when you do that they obey the vector space properties so the principle of superposition holds ie if |x1> and |x2> are any two pure states a linear combination is also a pure state. This is what is meant by a superposition. Note it deals with elements of a vector space not convex sums of pure states when considered operators - they are mixed states. This means the state 1/2 |x1> + 1/2 |x2> is a pure state and is totally different from the mixed state 1/2 |x1><x1| + 1/2 |x2><x2|.

Now to your question. Pure states, being an element of a vector space, can be expanded in terms of a basis. A wavefunction is simply the expansion in terms of the position basis. But that's just an arbitrary way of mathematically expressing it. It changes nothing - states are simply like probability - an aid to calculating the expected values of observations.

That's from the formalism - interpretations add their own take on it.

Thanks
Bill