Final problem-solving workshop

Gaussian-type integrals. These are the standard, quotable results for any Gaussian-shaped wavefunction such as a Gaussian wave packet $\psi(x)\propto e^{-x^2/2\sigma^2}$:

where: $a$ is a positive constant fixed by the physical width of the packet (for $\psi\propto e^{-x^2/2\sigma^2}$, $a=1/\sigma^2$ in $|\psi|^2$), and the third integral vanishes because $x\,e^{-ax^2}$ is an odd function integrated over a symmetric interval. These three results are the complete toolkit needed to normalize a Gaussian wavefunction and to compute $\langle x\rangle$ and $\langle x^2\rangle$ for it, without ever performing the integration by hand.

Sinusoidal-squared integrals. For the infinite-well wavefunction $\phi_n(x)=A\sin(n\pi x/L)$ on $[0,L]$, normalization requires:

where: $n$ is the quantum number and $L$ is the well width; the result is independent of $n$ because it follows from the half-angle identity $\sin^2\theta=\tfrac12(1-\cos2\theta)$ — the oscillating $\cos(2n\pi x/L)$ term completes a whole number of cycles over $[0,L]$ and integrates to exactly zero, leaving precisely half of $L$. The same half-angle trick governs time-averages of oscillatory quantities elsewhere in physics, e.g. $\langle\cos^2(\omega t)\rangle=\tfrac12$, which is why average power and intensity formulas (such as $I=\tfrac12 c\varepsilon_0 E_0^2$) always carry a factor of one-half.

Expectation values and symmetry. Once a wavefunction is normalized via $\int|\psi|^2dx=1$, expectation values take the form $\langle x\rangle=\int\psi^*x\,\psi\,dx$ and $\langle x^2\rangle=\int\psi^*x^2\,\psi\,dx$. For a real, symmetric probability density — such as $|\phi_n(x)|^2$ about the midpoint of a box, or $|\psi(x)|^2$ about the origin for a centred Gaussian — you rarely need to grind through the integral by hand: symmetry alone fixes $\langle x\rangle$, and only $\langle x^2\rangle$ needs the standard-integral machinery above. This division of labour — symmetry for the first moment, standard integrals for the second — is exactly the pattern examiners test: a quick qualitative argument for $\langle x\rangle$, followed by a short calculation for $\langle x^2\rangle$ using whichever of the two integral families matches the wavefunction's shape.

Choosing the right family. Before reaching for either toolkit, identify the wavefunction's shape and domain: a bell-shaped exponential extending over all space signals the Gaussian family, while a sine confined between hard walls on a finite interval signals the sinusoidal family. The two never mix within a single problem in this course — a wavefunction is either an unbounded Gaussian packet or a bounded box eigenstate, never both at once — so this identification step alone resolves most of the ambiguity about which formula to quote.

Normalize a Gaussian wave packet and find $\langle x^2\rangle$. A particle's wavefunction is $\psi(x)=Ne^{-x^2/2\sigma^2}$ with $\sigma=2.0\times10^{-10}\ \text{m}$. Find the normalization constant $N$ and the expectation value $\langle x^2\rangle$.

Step 1. Set up the normalization integral. Requiring $\int_{-\infty}^{\infty}|\psi|^2dx=1$ gives $N^2\int_{-\infty}^{\infty}e^{-x^2/\sigma^2}dx=N^2\sqrt{\pi\sigma^2}=1$, using the standard Gaussian result with $a=1/\sigma^2$.

Step 2. Solve for $N$. This gives $N=(\pi\sigma^2)^{-1/4}$. Numerically, $\sigma\sqrt\pi=(2.0\times10^{-10})(1.772)=3.545\times10^{-10}$, so $N\approx 5.31\times10^{4}\ \text{m}^{-1/2}$.

Step 3. Compute $\langle x^2\rangle$ using the second standard Gaussian integral: $\langle x^2\rangle=N^2\int_{-\infty}^{\infty}x^2e^{-x^2/\sigma^2}dx=N^2\cdot\dfrac{\sigma^2}{2}\sqrt{\pi\sigma^2}$. Since $N^2\sqrt{\pi\sigma^2}=1$ from Step 1, this simplifies immediately.

Step 4. Read off the answer: $\langle x^2\rangle=\dfrac{\sigma^2}{2}=\dfrac{(2.0\times10^{-10})^2}{2}=\mathbf{2.0\times10^{-20}\ m^2}$. The spread $\sqrt{\langle x^2\rangle}=\sigma/\sqrt2\approx1.4\times10^{-10}\ \text{m}$ is directly the $\Delta x$ that would enter the uncertainty relation $\Delta x\,\Delta p\geq\hbar/2$ from the wave-mechanics block.

Competitive stretch

The same packet's ground-state cousin, $\phi_1(x)=\sqrt{2/L}\sin(\pi x/L)$ on a box of width $L$, gives $\langle x\rangle=L/2$ purely by symmetry — no integration needed, since $|\phi_1(x)|^2$ is symmetric about the midpoint $x=L/2$ (check $|\phi_1(L-x)|^2=|\phi_1(x)|^2$ using $\sin(\pi-\theta)=\sin\theta$). Recognizing this shortcut in an exam saves the entire integration-by-parts calculation that a brute-force approach would require.

• Using $\int_{-\infty}^{\infty}e^{-ax^2}dx=\sqrt{\pi/a}$ or $\int_0^L\sin^2(n\pi x/L)dx=L/2$ outside their valid domain — the Gaussian result needs the full real line, the sine-squared result needs a full-period interval like $[0,L]$; applying $L/2$ to a partial interval such as $[0,L/2]$ gives the wrong answer because the cosine term no longer completes whole cycles.
• Grinding through integration by parts when a symmetry check would give $\langle x\rangle$ instantly — always check whether $|\psi|^2$ is even, odd, or symmetric about the interval's midpoint first, since a fully worked-out integral and the symmetry shortcut must agree, and the shortcut is far faster under exam conditions.
• Treating the normalization constant $N$ or $A$ as a free parameter chosen for convenience — it is fixed uniquely by requiring total probability equal one, and re-using a different value later in the same problem (say, to simplify an unrelated expression) silently breaks the normalization.
• Dropping the factor of one-half that comes from $\langle\sin^2\rangle=\langle\cos^2\rangle=\tfrac12$ when computing average power, intensity, or energy — a frequent slip under time pressure, and one that produces an answer exactly double the correct value.
• Forgetting that $\int_0^L\sin^2(n\pi x/L)dx=L/2$ holds for every quantum number $n$ — students sometimes assume the normalization constant $A=\sqrt{2/L}$ changes between the ground state and excited states, when in fact it is identical for all $n$.

Ask the tutor: want a few more practice problems mixing Gaussian and sinusoidal integrals, or a walkthrough applying this toolkit to a specific past-exam question?