Our solution to the quantum-mechanical problem of a particle in a box tells us that:

1. The particle must have energy $E_n=n^2{E}_1$, where $n=1\text{,}2\text{,}3\text{,}\dots$ is the quantum number. $E_1=h^2\text{/}8m{L}^2$ is the energy of the $n=1$ ground state.

2. The wave function for a particle in quantum state $n$ is

   $${\psi }_n\left(x\right)=\{\begin{array}{ll}\sqrt{{\displaystyle \frac{2}{L}}}\text{sin}\left({\displaystyle \frac{n\pi x}{L}}\right)\hfill & 0\le x\le L\hfill \\ 0\hfill & x<0\text{ and }x>L\hfill \end{array}$$

   These are the stationary states of the system.

3. The probability density for finding the particle at position $x$ inside the box is

   $$P_n\left(x\right)=\left|{\psi }_n\left(x\right)\right|{}^2={\displaystyle \frac{2}{L}}{\text{sin}}^2\left({\displaystyle \frac{n\pi x}{L}}\right)$$ (40.28)

A graphical presentation will make these results more meaningful. FIGURE 40.7 shows the wave functions $\psi \left(x\right)$ and the probability densities $P\left(x\right)=\left|\psi \left(x\right)\right|{}^2$ for quantum states $n=1$ to 3. Notice that the wave functions go to zero at the boundaries and thus are continuous with $\psi =0$ outside the box.

FIGURE 40.7 Wave functions and probability densities for a particle in a rigid box of length $L$.

The wave functions $\psi \left(x\right)$ for a particle in a rigid box are analogous to standing waves on a string that is tied at both ends. You can see that ${\psi }_n(x)$ has $(\mathit{n}-1)$ nodes (zeros), excluding the ends, and $\mathit{n}$ antinodes (maxima and minima). This is a general result for any wave function, not just for a particle in a rigid box.

FIGURE 40.8 shows another way in which energies and wave functions are shown graphically in quantum mechanics. First, the graph shows the potential-energy function $U\left(x\right)$ of the particle. Second, the allowed energies are shown as horizontal lines (total energy lines) across the potential-energy graph. These are labeled with the quantum number $n$ and the energy $E_n$. Third—and this is a bit tricky—the wave function for each $n$ is drawn as if the energy line were the zero of the $y$-axis. That is, the graph of ${\psi }_n\left(x\right)$ is drawn on top of the $E_n$ energy line. This allows energies and wave functions to be displayed simultaneously, but it does not imply that ${\psi }_2$ is in any sense “above” ${\psi }_1$. Both oscillate sinusoidally about zero, as Figure 40.7 shows.

FIGURE 40.8 An alternative way to show the potential-energy diagram, the energies, and the wave functions.

EXAMPLE 40.2 Energy levels and quantum jumps

A semiconductor device known as a quantum-well device is designed to “trap” electrons in a 1.00-nm-wide region. Treat this as a one-dimensional problem.

1. What are the energies of the first three quantum states?

2. What wavelengths of light can these electrons absorb?

MODEL Model an electron in a quantum-well device as a particle confined in a rigid box of length $L=1.00\text{nm}$.

VISUALIZE FIGURE 40.9 shows the first three energy levels and the transitions by which an electron in the ground state can absorb a photon.

FIGURE 40.9 Energy levels and quantum jumps for an electron in a quantum-well device.

FIGURE 40.9 Full Alternative Text

SOLVE

1. The particle’s mass is $m=m_{\text{e}}=9.11\times {10}^{-31}\text{kg}$. The allowed energies, in both J and eV, are

   $$\begin{array}{lll}\hfill E_1& =\hfill & {\displaystyle \frac{h^2}{8m{L}^2}}=6.03\times {10}^{-20}\text{ J}=0.377\text{ eV}\hfill \\ \hfill E_2& =\hfill & 4E_1=1.508\text{ eV}\hfill \\ \hfill E_3& =\hfill & 9E_1=3.393\text{ eV}\hfill \end{array}$$

2. An electron spends most of its time in the $n=1$ ground state. According to Bohr’s model of stationary states, the electron can absorb a photon of light and undergo a transition, or quantum jump, to $n=2$ or $n=3$ if the light has frequency $f=\text{Δ}E\text{/}h$. The wavelengths, given by $\lambda =c\text{/}f=hc\text{/}\text{Δ}E$, are

   $$\begin{array}{lll}\hfill {\lambda }_{1\to 2}& =\hfill & {\displaystyle \frac{hc}{E_2-E_1}}=1098\text{ nm }\hfill \\ \hfill {\lambda }_{1\to 3}& =\hfill & {\displaystyle \frac{hc}{E_3-E_1}}=411\text{ nm}\hfill \end{array}$$

REVIEW In practice, various complications usually make the $1\to 3$ transition unobservable. But quantum-well devices do indeed exhibit strong absorption and emission at the $\lambda {}_{1\to 2}$ wavelength. In this example, which is typical of quantum-well devices, the wavelength is in the near-infrared portion of the spectrum. Devices such as these are used to construct the semiconductor lasers used in DVD players and laser printers.