39-2 Wave Functions of a Trapped Electron

39-2 Wave Functions of a Trapped Electron#

Prompts

  • For an electron in a one-dimensional infinite potential well of length \(L\), write the wave function \(\psi_n(x)\) for quantum number \(n\). What is the normalized form? (Hint: \(\int_0^L \psi_n^2\,dx = 1\).)

  • Define probability density \(|\psi_n|^2\). What does \(|\psi_n(x)|^2\,dx\) represent? How do you find the probability of detecting the electron between \(x_1\) and \(x_2\)?

  • For \(n=2\), where is the electron most likely to be found? Where is it least likely? Sketch \(|\psi_n|^2\) for \(n=1\) and \(n=2\).

  • State the correspondence principle. As \(n\) increases, what happens to the probability density across the well?

  • Normalize the wave function \(\psi_n = A\sin(n\pi x/L)\): find \(A\) such that \(\int_0^L \psi_n^2\,dx = 1\).

Lecture Notes#

Overview#

  • Solving Schrödinger’s equation for an electron in a one-dimensional infinite potential well yields wave functions \(\psi_n(x) = \sqrt{2/L}\,\sin(n\pi x/L)\) for \(0 < x < L\).

  • The probability density \(|\psi_n|^2\) gives the probability per unit length of detecting the electron. The probability of detection between \(x_1\) and \(x_2\) is \(\int_{x_1}^{x_2} |\psi_n|^2\,dx\).

  • Normalization requires \(\int_{-\infty}^{\infty} |\psi_n|^2\,dx = 1\); this fixes the amplitude \(A = \sqrt{2/L}\).

  • The correspondence principle: at large quantum numbers \(n\), quantum predictions merge with classical ones (e.g., probability becomes more uniform across the well).

Wave functions for the infinite well#

Solving Schrödinger’s equation for an electron in a one-dimensional infinite potential well (section 39-1) with boundary conditions \(\psi = 0\) at \(x = 0\) and \(x = L\) yields

(388)#\[ \psi_n(x) = A\sin\left(\frac{n\pi x}{L}\right), \quad n = 1, 2, 3, \ldots \]

for \(0 < x < L\); \(\psi_n = 0\) outside. This has the same form as the displacement of a standing wave on a string with fixed ends—the electron is a standing matter wave.

Normalization requires that the total probability of finding the electron somewhere is 1:

(389)#\[ \int_{-\infty}^{\infty} |\psi_n|^2\,dx = \int_0^L \psi_n^2\,dx = 1 \]

Evaluating the integral gives \(A = \sqrt{2/L}\). The normalized wave functions are

(390)#\[ \psi_n(x) = \sqrt{\frac{2}{L}}\,\sin\left(\frac{n\pi x}{L}\right), \quad 0 < x < L \]

Probability density and detection probability#

The probability density \(|\psi_n(x)|^2\) is the probability per unit length of detecting the electron at position \(x\). For the infinite well, \(\psi_n\) is real, so \(|\psi_n|^2 = \psi_n^2\):

(391)#\[ |\psi_n(x)|^2 = \frac{2}{L}\sin^2\left(\frac{n\pi x}{L}\right), \quad 0 < x < L \]

The probability of detecting the electron between \(x_1\) and \(x_2\) is

(392)#\[ P(x_1 \leq x \leq x_2) = \int_{x_1}^{x_2} |\psi_n(x)|^2\,dx \]

  • For \(n = 1\) (ground state): maximum at center (\(x = L/2\)); zero at walls.

  • For \(n = 2\): two maxima near \(x = L/4\) and \(3L/4\); zero at \(x = 0\), \(L/2\), \(L\).

  • As \(n\) increases, the number of maxima increases; the probability becomes more evenly distributed.

Physical meaning

We cannot “see” the wave function. We can only probe for the electron; each detection finds it at a point. Repeated detections build up a distribution matching \(|\psi_n|^2\).

Correspondence principle#

For large quantum number \(n\), the probability density \(|\psi_n|^2\) has many oscillations across the well and becomes nearly uniform. Correspondence principle (Bohr): at large enough \(n\), quantum predictions merge with classical physics. Classically, we would expect the electron to be equally likely anywhere in the well; quantum mechanics gives that result in the limit of large \(n\).

Example

Ground state, left one-third: For \(n=1\) and \(L=100\) pm, the probability of detection in \(0 \leq x \leq L/3\) is \(\int_0^{L/3} (2/L)\sin^2(\pi x/L)\,dx\). With \(y = \pi x/L\), this becomes \((2/\pi)\int_0^{\pi/3}\sin^2 y\,dy = 2/\pi \cdot (y/2 - \sin(2y)/4)\big|_0^{\pi/3} \approx 0.20\). So about 20% of detections fall in the left third. By symmetry, 20% in the right third; 60% in the middle third.

Summary#

  • Wave functions \(\psi_n(x) = \sqrt{2/L}\,\sin(n\pi x/L)\) for \(0 < x < L\); zero outside.

  • Probability density \(|\psi_n|^2 = (2/L)\sin^2(n\pi x/L)\); probability between \(x_1\) and \(x_2\) is \(\int_{x_1}^{x_2} |\psi_n|^2\,dx\).

  • Normalization \(\int |\psi_n|^2\,dx = 1\) fixes \(A = \sqrt{2/L}\).

  • Correspondence principle: large \(n\) \(\Rightarrow\) probability approaches uniform (classical).