2.2. Calculating observable quantities from the wave function

The wave function of a particle gives us the probability density function that contains information about all of the possible observable quantities for that particle. If we wish to measure the position of a particle we can perform a measurement. Now imagine we are able to set up the same experiment with the same conditions and repeat the measurement, this time we will most likely get a different result. That is in the nature of the wave function. We can no longer predict the outcome as you can with a classical object but we can calculate the expected value - the average seen over multiple measurements of an object given the same initial conditions.

2.2.1. Expectation values

The wave function squared gives the probability density function for a particle. That can be interpreted as the probability per unit distance for observing the particle. We will need to extract other information from the wave function such at the momentum and energy.

As we saw in the last section, when we try to localise a particle the wave function is still distributed over a range of \(x\) values. If we wish to work out the position of a particle then it is not possible to obtain a single value so we must use the appropriate probability density function and obtain the average position found if we perform multiple experiments on the same system. The expectation value for position is given as

\[ <x>=\int_{-\infty}^{\infty}x P(x) = \int_{-\infty}^{\infty} \psi^*(x) x \psi(x) dx \]

where \(<x>\) is the expectation value of position and \(x\) is the position operator. The position operator for a single dimension in Cartesian coordinates is quite simple. To perform the same for momentum will require us to find the momentum operator. In general we express the expectation value as

\[ <O>=\int_{-\infty}^{\infty}\psi^*(x) \hat{O}\psi(x) dx \]

where \(O\) is the expectation value of some property we are interested in observing and \(\hat{O}\) is the associated operator that creates the appropriate function from the wave function.

Definition of expectation value

The expectation value of an operator is the average value of the observable property represented by the operator \(\hat{O}\) obtained after a large number of repeated measurements of the property using identical systems.

2.2.2. Operators

In order to extract information from the wave function about momentum and energy we will need operators for each quantity. We know from the de Broglie relationship that

\[ p=\hbar k \]

Let’s define a wave function with momentum \(\hbar k\).

\[ \psi(x)=Ae^{ik x} \]

differentiating this function acts in such a way that the \(k\) is brought to the front of the function.

\[ \frac{d}{dx}\psi(x)=ik Ae^{ikx} \]

multiply through by \(\frac{\hbar}{i}\)

\[ \frac{\hbar}{i}\frac{d}{dx}\psi(x)=\hbar kAe^{ikx} \]

in other words, allowing the operator \(\frac{\hbar}{i}\frac{d}{dx}\) to act on the wave function brings out a factor of \(\hbar k\) which is of course the momentum of the particle we initially used. We now make the assumption that this operator is related to the momentum of the particle. If we wish to find the expectation value for the momentum of a particle defined by the wave function \(\psi(x)\) we evaluate

\[ <p>=\int_{-\infty}^{\infty}\psi^*(x)\hat{p}\psi(x) dx \]

Other operators that will be useful in this course.

Dynamical Quantity	Operator
Momentum	\(\hat{p}=-i\hbar\frac{d}{dx}=\frac{hbar}{i}\frac{d}{dx}\)
Kinetic Energy	\(\hat{E_k}= -\frac{\hbar^2}{2m}\frac{d^2}{dx^2}\)
Total Energy	\(\hat{E}=i\hbar\frac{d}{dt}\)
Position	\(\hat{x}=x\)
Potential Energy	\(\hat{V}=V{(x)}\)