# What is a wave function

1. Jul 24, 2014

### Greg Bernhardt

Definition/Summary

A wave function is a mathematical function that describes a physical system in quantum mechanics. The time evolution of this wave function, and thus of the system itself, is described by the Schrodinger equation.

Equations

$$P_G=\int_{G}\psi^*\psi d^3 x$$

Schrodinger Equation:

$$\frac{-\hbar^2}{2m} \nabla^2 \psi +V\psi =i\hbar\frac{\partial\psi}{\partial t}$$

Extended explanation

Basic Concept

The wavefunction that describes a quantum mechanical system contains all the information that can be known about that system. The wavefunction itself is not observable. In fact, it can be complex-valued. However, the absolute square of the wavefunction is observable. The absolute square of the wavefunction is a probability density.

For example, if the wavefunction is expressed in real space and our system is a particle, the absolute square gives a probability density for the position of the system. Integrating this probability density between some bounds will give the probability that the particle will be found in that region when its position is measured. For a volume G:

$$P_G=\int_{G}\psi^*\psi d^3 x$$

Similarly, if your wave function is expressed in momentum space, the probability density will be a probability density for the momentum of the particle.

There are several properties every physically possible wavefunction must have:

1)The function must be continuous.

2)The absolute square of the function must be normalizable. (If it is not normalizable, then it cannot possible represent any real probability density.)

The wave function can also be used to find the expectation value of any observable of the system. This is done by placing the operator corresponding to the observable in between the wavefunction and it's conjugate inside the integral. For example:

Position:

$$<x>=\int_a^b \psi^*x\psi dx$$

Momentum:

$$<p>=\int_c^d \psi^*(\frac{h}{i}\frac{d}{dx})\psi dx$$

Generalization

In the general formalization of quantum mechanics wave functions are represented as vectors (either finite dimensional or infinite dimensional) in a complex Hilbert space.

The normalization condition for these vectors is $<\psi|\psi> = 1$ where the inner product is defined as:

$$<f|g>=\int_V f^*g dV$$

In this formalism, observables are represented by hermitian operators. i.e. the hermitian operator, A, has the property that $A=A^+$.

Determinate states (states which are guaranteed to return the same value for an observable with operator $\hat{A}$) are eigenstates of that operator. In other words, these determinate states are solutions to the following equation:

$$A\psi=a\psi$$ where $\psi$ is the determinate state, and a, the eigenvalue, is the value any measurement of the observable is guaranteed to return.

An important property of the eigenstates of an operator is that they are complete. This means that any other vector in the Hilbert space, i.e. any other possible state for the system, can be written as a linear combination of the basis states.

Example
A quick example of the completeness of eigenstates is in regards to a spin 1/2 system. An electron with spin, s=1/2 has two spin eigenstates, spin up, $|\uparrow >$ and spin down, $|\downarrow >$. Any other spin state the system can be placed in can then be expressed as:

$$|s>=a|\uparrow > + b|\downarrow >$$

where $a^2$ & $b^2$ are the probabilities that a measurement of the spin of the system will return a measurement of spin up or spin down, respectively.

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