1.10: Wave packets and uncertainty

We now have two ways to describe an electron. We could describe it as a plane wave, with precisely defined wavenumber and angular frequency:

\[ \psi(x,t) = e^{i(k_{0}x-\omega_{0}t)} \nonumber \]

But as we have seen, the intensity/probability density of the plane wave is uniform over all space (and all time). Thus, the position of the electron is perfectly uncertain – it is probability distribution is uniform everywhere in the entire universe. Consequently, a plane wave is not usually a good description for an electron.

On the other hand, we could describe the electron as an idealized point particle

\[ \psi(x,t)=\delta(x-x_{0},t-t_{0}) \nonumber \]

existing at a precisely defined position and time. But the probability density of the point particle is uniform over all of k-space and the frequency domain. We will see in the next section that this means the energy and momentum of the electron is perfectly uncertain, i.e. arbitrarily large electron energies and momenta are possible.

The only alternative is to accept an imprecise description of the electron in both real space and k-space, time and the frequency domain. A localized oscillation in both representations is called a wave packet. A common wavepacket shape is the Gaussian. For example, instead of the delta function, we could describe the electron's position as

\[ \psi(x) = a\text{ exp}[-\frac{1}{4}(\frac{x}{\sigma_{x}})^{2}] \nonumber \]

This function was chosen such that the probability distribution of the electron is

Figure \(\PageIndex{1}\): A normalized Gaussian. The Gaussian has the same shape in real space and k-space. Note that a Gaussian approximates a delta function in the limit \(\sigma_{x} \rightarrow 0\).

\[ |\psi(x)|^{2} = |a|^{2} \text{ exp}[-\frac{1}{2}(\frac{x}{\sigma_{x}})^{2}] \nonumber \]

where \(\sigma_{x}\) is the standard deviation. \(\sigma_{x}\) measures the width of the Gaussian and is often thought of as the uncertainty in the position of the electron. The constant, a, is determined by normalizing the probability density over all space, i.e., integrating Equation (1.10.4) over x, we get

\[ a=(2\pi \sigma_{x}^{2})^{-1/4} \nonumber \]

Strictly, the uncertainty of a given quantity is defined by

\[ \sigma^{2} = \langle(x-\langle x\rangle)^{2}\rangle \nonumber \]

\( = \langle x^{2}-2x\langle x\rangle+\langle x\rangle^{2}\rangle \)

where \(\langle x\rangle\) signifies the average or expectation value of x. Because \(\langle x\rangle\) is a constant Equation (1.10.6) may be simplified:

\[ \sigma^{2}= \langle x^{2}\rangle-2\langle x \rangle \langle x\rangle+\langle x\rangle^{2} \nonumber \]

\( = \langle x^{2}\rangle-\langle x\rangle^{2} \)

We will leave it as an exercise to show that for the Gaussian probability density:

\[ \sigma^{2} = \int x^{2} |\psi(x)|^{2}dx = \sigma_{x}^{2} \nonumber \]

Thus, the Gaussian is a convenient choice to describe a wavepacket because it has a readily defined uncertainty.

In k-space, the electron is also described by a Gaussian (this is another of the convenient properties of this function). Application of the Fourier transform in Equation (1.9.2) and some algebra gives

\[ A(k)=(8\pi\sigma_{x}^{2})^{1/4}\text{ exp}[-\sigma_{x}^{2}k^{2}] \nonumber \]

The probability distribution in k-space is

\[ |A(k)|^{2}=(8\pi\sigma_{x}^{2})^{1/2}\text{ exp}[-2\sigma_{x}^{2}k^{2}] \nonumber \]

Thus, the uncertainty in k-space is

\[ \frac{1}{2\sigma_{k}^{2}} = 2\sigma_{x}^{2} \nonumber \]

The product of the uncertainties in real and k-space is

\[ |\sigma_{x}\sigma_{k}| = \frac{1}{2} \nonumber \]

The product \(|\sigma_{x}||\sigma_{k}| \geq |\sigma_{x}\sigma_{k}|\). Thus,

\[ |\sigma_{x}||\sigma_{k}| \geq \frac{1}{2} \nonumber \]