Diagnostic Checklist

Try the problems relevant to what you want to study next. Each tests a small calculation, gives a solution, and points to an explanation if you need one. You do not need to solve every problem before starting quantum mechanics.

Helpful background. notation for reading the symbols used below.

Exercises in vectors, wavefunctions, and numerical accuracy

Vectors and probabilities

These first two checks are enough to identify common gaps before studying finite quantum systems.

Normalize a complex vector. For $v=(1,2i)^T$, compute $v^\dagger v$ and a normalized vector in the same direction.

Solution

The adjoint is $v^\dagger=(1,-2i)$, so $v^\dagger v=1+(-2i)(2i)=5$. The normalized vector is $v/\sqrt5$. Using a transpose without complex conjugation would incorrectly give $-3$.

Review Inner Products and Complex Vector Spaces.

Project onto an outcome. Let $|\psi\rangle=(1,2i)^T/\sqrt5$ and $P=\operatorname{diag}(1,0)$. Check $P^2=P=P^\dagger$ and compute $\langle\psi|P|\psi\rangle$.

More generally, let $P_1,P_2$ be orthogonal projectors with $P_1+P_2=I$. Show that their Born probabilities sum to one in a normalized state.

Solution

Squaring or taking the adjoint leaves this diagonal matrix unchanged. Also $P|\psi\rangle=(1,0)^T/\sqrt5$, so the expectation is $1/5$. The complementary projector gives $4/5$.

For the general pair, linearity and completeness give

$$p_1+p_2=\langle\psi|(P_1+P_2)|\psi\rangle =\langle\psi|\psi\rangle=1.$$

The projector identity supplies the mathematics; the Born rule interprets these expectation values as outcome probabilities.

Review Projectors and the Born Rule. For an ordered route through the basics, use Math Needed for Core Formalism.

Wavefunctions and boundary conditions

Normalize a function. On $0<x<L$, let $\psi(x)=A\sin(\pi x/L)$ and take $A$ real and positive. Find $A$.

Solution

Since $\int_0^L\sin^2(\pi x/L)dx=L/2$, normalization requires $A^2L/2=1$. Thus $A=\sqrt{2/L}$, with units of length to the power $-1/2$.

Review L² Spaces if the integral or units are unfamiliar.

Find the allowed wave numbers. Solve $\psi''+k^2\psi=0$ on $[0,L]$ for $k>0$ with $\psi(0)=\psi(L)=0$ and $\psi\ne0$.

Solution

Write $\psi=A\sin(kx)+B\cos(kx)$. The first boundary condition sets $B=0$; the second requires $\sin(kL)=0$. Thus $k=n\pi/L$ for positive integer $n$. At $k=0$ the solution is linear and the two boundary conditions force it to vanish, so it supplies no additional state.

Review Ordinary Differential Equations and Boundary Conditions. Continue with the wave-mechanics guide.

Fourier transforms and distributions

Check a momentum scale. A Gaussian packet has position standard deviation $\sigma_x$ and saturates $\sigma_x\sigma_p=\hbar/2$. What happens to $\sigma_p$ if $\sigma_x$ is halved?

Solution

Its momentum spread doubles. This statement applies to the minimum-uncertainty Gaussian specified in the question; arbitrary states obey an inequality and need not saturate it.

Review Wave Packets for the calculation and Fourier Transform for conventions.

Use a delta function. For smooth $f$, evaluate $\int_{-\infty}^{\infty}f(x)\delta(2x-4)dx$.

Solution

The change of variable $u=2x-4$ gives $dx=du/2$, so the result is $f(2)/2$. The scale factor matters: $\delta(2x-4)=\delta(x-2)/2$.

Review Delta Function if you obtained $f(2)$.

Probability and spin

Distinguish a density from a probability. Let $X$ be uniform on $[0,1/4]$. What is its density, and what is $P(X\le1/8)$?

Solution

The density is $4$ on that interval and zero elsewhere. The probability is the area $4(1/8)=1/2$. A density may exceed one; its integral over an event must lie between zero and one.

Review Probability Densities.

Check noncommuting matrices. Compute $[\sigma_x,\sigma_y]$ using the Pauli matrices.

Solution

Matrix multiplication gives $\sigma_x\sigma_y=i\sigma_z$ and $\sigma_y\sigma_x=-i\sigma_z$. Their difference is $2i\sigma_z$. The matrices do not commute. In this particular example, however, the two products differ only by a global minus sign, so applying either full sequence to a state gives the same final measurement probabilities. Noncommuting matrices need not give experimentally distinguishable sequences in every special case.

Review Pauli Matrices, then follow the spin and symmetry guide.

Numerical accuracy

Interpret refinement. An energy estimate changes by $0.04$ when a grid spacing is halved, and by $0.01$ on the next halving. Is this consistent with a leading error proportional to the square of the grid spacing?

Solution

Yes. If $E(h)=E_*+Ch^2$ to leading order, successive differences decrease by a factor of four. Under that model, the remaining error at the finest grid is approximately $0.01/3$. More refinements and checks of other error sources are needed before trusting the estimate.

Review Convergence Tests and Error Estimates.

Choose the next step

If you can do a calculation but forget its notation, use Mathematical Notation. If a solution contains an unfamiliar idea, read its linked explanation and try the problem again. Advanced topics such as operator domains, tensor products, and geometric phase can be learned when your chosen physics route reaches them.

References

• S. Axler, Linear Algebra Done Right, 3rd ed., Springer, 2015 — the vector and operator calculations. Publisher.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994 — wavefunction normalization, Fourier methods, and spin.
• L. N. Trefethen and D. Bau III, Numerical Linear Algebra, SIAM, 1997 — interpreting numerical approximations and errors.