7.1 Wavefunctions and Probability

With a normalized wavefunction^7.1

$\begin{displaymath} \int^{\infty}_{-\infty} \vert\Psi(x,t)\vert^2 dx = 1, \end{displaymath}$ (7.1)

the probability density

$\begin{displaymath} \varrho(x,t) = \Psi^*(x,t) \Psi(x,t) = \vert\Psi(x,t)\vert^2 \end{displaymath}$ (7.2)

describes the probability distribution of the particle or particles described by $\Psi$ .
The square of the wavefunction of a classical wave is proportional to its energy density and intensity (see Section's 4.2 and 5.4).
- In the quantum mechanical context, probability density takes the place of energy density.
- The quantum analog of intensity is probability current introduced in Section 7.3.
Why do we take the complex square of quantum mechanical wavefunctions and the square of the real part of classical ones?
- Classical wavefunctions must be real functions. They describe measurable quantities. We sometimes use complex exponentials for mathematical convenience, but we use only the real part to describe the physical system.
- Quantum mechanical wavefunctions are complex functions. The complex square gives the magnitude (squared) of a complex function. Taking the real part would throw away an important part of the wavefunction.
The probability density can be used to calculate the probability of finding the system in a particular position interval

$\begin{displaymath} P_{ab}(t) = \int_a^b \varrho(x,t) dx \end{displaymath}$ (7.3)
An operator is associated with each observable quantity. Examples for one dimensional systems include position

$\begin{displaymath} \hat{x} = x, \end{displaymath}$ (7.4)

momentum

$\begin{displaymath} \hat{p} = -i\hbar \frac{\partial}{\partial x}, \end{displaymath}$ (7.5)

and kinetic energy

$\begin{displaymath} \hat{K} = \frac{\hat{p}^2}{2m} = -\frac{\hbar^2}{2m} \frac{\partial^2}{\partial x^2} \end{displaymath}$ (7.6)
The expectation value of an operator is a kind of weighted average^7.2

$\begin{displaymath} \left< o \right> = \int^\infty_{-\infty} \Psi(x,t)^* \hat{O} \Psi(x,t) dx. \end{displaymath}$ (7.7)

giving the average value of repeated measurements of made on identical systems. Expectation values can be time dependent.
A word about units : Eq. 7.1 implies

$\begin{displaymath}[\Psi]= \frac{1}{\sqrt{\mathrm{L}}} \end{displaymath}$

and hence

$\begin{displaymath}[\varrho]= \frac{1}{\mathrm{L}} \end{displaymath}$

for one dimensional systems. Hence the units of the expectation value $\left< o \right>$ are simply the units of the quantity .

Next: 7.2 The Schrödinger Equation Up: 7. Quantum Mechanical Waves Previous: 7. Quantum Mechanical Waves