7.1 Wave functions  (Page 3/22)

Strategy

Solution

where A is the amplitude of the wave function and $k=2\text{π}\text{/}\lambda$ is its wave number. Beyond this interval, the amplitude of the wave function is zero because the ball is confined to the tube. Requiring the wave function to terminate at the right end of the tube gives

Evaluating the wave function at $x=L\text{/}2$ gives

This equation is satisfied if the argument of the cosine is an integral multiple of $\pi \text{/}2,3\pi \text{/}2,5\pi \text{/}2,$ and so on. In this case, we have

or

Applying the normalization condition gives $A=\sqrt{2\text{/}L}$ , so the wave function of the ball is

To determine the probability of finding the ball in the last quarter of the tube, we square the function and integrate:

Significance

This momentum is much too small to be measured by any human instrument.

An interpretation of the wave function

We are now in position to begin to answer the questions posed at the beginning of this section. First, for a traveling particle described by $\text{Ψ}\left(x,t\right)=A\text{sin}(kx-\omega t)$ , what is “waving?” Based on the above discussion, the answer is a mathematical function that can, among other things, be used to determine where the particle is likely to be when a position measurement is performed. Second, how is the wave function used to make predictions? If it is necessary to find the probability that a particle will be found in a certain interval, square the wave function and integrate over the interval of interest. Soon, you will learn soon that the wave function can be used to make many other kinds of predictions, as well.

Third, if a matter wave is given by the wave function $\text{Ψ}(x,t)$ , where exactly is the particle? Two answers exist: (1) when the observer is not looking (or the particle is not being otherwise detected), the particle is everywhere $(x=\text{−}\infty ,\text{+}\infty )$ ; and (2) when the observer is looking (the particle is being detected), the particle “jumps into” a particular position state $(x,x+dx)$ with a probability given by $P(x,x+dx)=|\text{Ψ}(x,t){|}^{2}dx$ —a process called state reduction    or wave function collapse    . This answer is called the Copenhagen interpretation    of the wave function, or of quantum mechanics.