Question

The diagram shows the Fourier transform $\mathit{A}\mathbf{\left(}\mathit{k}\mathbf{\right)}$of a Gaussian wave function$\mathit{\psi }\mathbf{\left(}\mathit{x}\mathbf{\right)}$that represents a reasonably well-localized particle.

(a) Determine approximate quantitative values for the wave function's wavelength and for the particle's position uncertainty.

(b) Can you determine the particle's approximate position? Why of why not?

Short answer

(a). The approximate quantitative value for thewavelengthand for the particle’s position uncertainty is$\Delta x=0.5\text{nm}$.

(b). No,the position itself is inconclusive.

Step-by-step solution

Given Data

The quantitative value of the wavelength$k=20{\text{nm}}^{-1}$

Concept of the wavelength

Formula used:

$\mathit{k}\mathbf{=}\frac{\mathbf{2}\mathbf{\pi }}{\mathbf{\lambda }}$,

Where$\mathit{k}$ is quantitative value of the wavelength, $\mathit{\lambda }$is the wavelength.

Step 3: Determine approximate quantitative values for the wave function's wavelength 

(a)

The quantitative value of the wavelength will correspond to the high contribution coming from the central value at$k=20{\text{nm}}^{-1}$.

Hence, the wavelength will be given as follows.

Substitute the value,

$\lambda =\frac{2\pi }{20\times {10}^9{\text{m}}^{-1}}=0.314\text{nm}$

For the position uncertainty, use uncertainty relation for the Fourier transformation (with $k$instead of$p$}

$\begin{array}{c}\Delta x\Delta k=\frac{1}{2}\\ \Delta x=\frac{1}{2\Delta k}\Delta x\\ =\frac{1}{2\times I_{{\text{nm}}^{-1}}}\Delta x\\ =0.5\text{nm}\end{array}$

Where the value of$\Delta k$ could be read from the graph, corresponding to the width of the curve at$\text{1/e}$ of the peak value, it tunes out to be approximately.

Determine the particle approximate position 

(b)

Unfortunately, the particle is represented as a wave, hence the wave loses its particle nature in the first place there. Nevertheless, if it is transformed to the k-space, it is only the wave number that is not sufficient to give the approximate position of the particle. At most it is possible to find the certainty in position as in part a, however, the position itself is inconclusive.