7.1 Wave functions  (Page 5/22)

where ${\text{Ψ}}^{*}\left(x,t\right)$ is the complex conjugate of the wave function. The complex conjugate of a function is obtaining by replacing every occurrence of $i=\sqrt{\mathrm{-1}}$ in that function with $\text{−}i$ . This procedure eliminates complex numbers in all predictions because the product ${\text{Ψ}}^{*}\left(x,t\right)\text{Ψ}\left(x,t\right)$ is always a real number.

Consider the motion of a free particle that moves along the x -direction. As the name suggests, a free particle experiences no forces and so moves with a constant velocity. As we will see in a later section of this chapter, a formal quantum mechanical treatment of a free particle indicates that its wave function has real and complex parts. In particular, the wave function is given by

where A is the amplitude, k is the wave number, and $\omega$ is the angular frequency. Using Euler’s formula, $e^{i\varphi }=\text{cos}(\varphi )+i\text{sin}\left(\varphi \right),$ this equation can be written in the form

where $\varphi$ is the phase angle. If the wave function varies slowly over the interval $\text{Δ}x,$ the probability of finding the particle in that interval is

If A has real and complex parts $(a+ib$ , where a and b are real constants), then

Notice that the complex numbers have vanished. Thus,

is a real quantity. The interpretation of ${\text{Ψ}}^{*}\left(x,t\right)\text{Ψ}\left(x,t\right)$ as a probability density ensures that the predictions of quantum mechanics can be checked in the “real world.”

Expectation values

In classical mechanics, the solution to an equation of motion is a function of a measurable quantity, such as x ( t ), where x is the position and t is the time. Note that the particle has one value of position for any time t . In quantum mechanics, however, the solution to an equation of motion is a wave function, $\text{Ψ}\left(x,t\right).$ The particle has many values of position for any time t , and only the probability density of finding the particle, $|\text{Ψ}(x,t){|}^2$ , can be known. The average value of position for a large number of particles with the same wave function is expected to be

This is called the expectation value    of the position. It is usually written

where the x is sandwiched between the wave functions. The reason for this will become apparent soon. Formally, x is called the position operator    .

At this point, it is important to stress that a wave function can be written in terms of other quantities as well, such as velocity ( v ), momentum ( p ), and kinetic energy ( K ). The expectation value of momentum, for example, can be written

Where dp is used instead of dx to indicate an infinitesimal interval in momentum. In some cases, we know the wave function in position, $\text{Ψ}\left(x,t\right),$ but seek the expectation of momentum. The procedure for doing this is