Types of Waves — Question 6

Question 6

A probability wave is not a hidden material rope

A particle is described by a complex amplitude $\psi(x,t)$. You are supplied the rule that detection probability in a small interval $\Delta x$ is approximately $|\psi|^2\Delta x$. Two preparations have local amplitude magnitudes in the ratio $2:1$, with equal interval widths.

1. Find the ratio of the local detection probabilities, assuming each full preparation is properly normalized.

2. Explain why a plotted real part of $\psi$ cannot be interpreted as a literal transverse displacement of matter. Does a zero real part force zero probability?

Question 6 - Solution

A probability wave is not a hidden material rope

Method and mathematical working

Apply the supplied measurement rule to each independently normalized preparation. The relevant quantity is the magnitude squared, which includes both real and imaginary components.$$\begin{aligned}|\psi|^2&=(\operatorname{Re}\psi)^2+(\operatorname{Im}\psi)^2,\\ \frac{P_1}{P_2}&=\frac{|\psi_1|^2\Delta x}{|\psi_2|^2\Delta x}.\\[4pt]\frac{P_1}{P_2}&=\left(\frac21\right)^2=4,\qquad \psi=ib\Rightarrow\operatorname{Re}\psi=0,\quad |\psi|^2=b^2.\end{aligned}$$

(a) Calculation and conclusion

The probability-density ratio is the squared magnitude ratio, $2^2:1^2=4:1$. Equal interval widths preserve that ratio. Proper normalization is required to interpret the amplitudes quantitatively.

(b) Interpretation and limits

The supplied observable depends on the complex magnitude, not a physical displacement of an elastic medium. A purely imaginary nonzero amplitude has zero real part but nonzero $|\psi|^2$. The wave description therefore differs from a rope-displacement field.

Physical check

For $\psi=ib$ with real nonzero $b$, the real part vanishes but the probability density is $b^2$. This provides an explicit counterexample to interpreting the plotted real part as the detection probability.