1. Mathematical foundations: amplitudes, bases and units
Position in the course: Lesson 1 of 10. Complete the preceding derivation and use the explained exercises to check understanding.
Prerequisites: Algebra, derivatives and elementary integration
1. Why complex vectors enter physics
A probability is a nonnegative real number; a probability amplitude may be complex. Adding amplitudes before squaring permits interference, whereas adding probabilities describes an incoherent alternative. Write \(z=a+ib\), \(z^*=a-ib\) and \(|z|^2=a^2+b^2\). A phase \(e^{i\theta}\) has unit modulus. Multiplying every amplitude by the same phase cannot change a prediction, but changing one component's phase can. Complex numbers therefore encode experimentally meaningful relative phases without making measured quantities complex.
For column vectors use the Hermitian inner product \(u^\dagger v\), with conjugation on the first factor. A normalized vector satisfies \(v^\dagger v=1\). In a function space this becomes \(\int\psi^*(x)\psi(x)dx=1\). The integration measure is part of the definition: in spherical coordinates it contains \(r^2\sin\theta\), so a radial function cannot be normalized with the Cartesian measure by accident.
2. Basis changes preserve the state
Let the columns of \(U\) be an orthonormal new basis, so \(U^\dagger U=I\). The old and new coefficient columns satisfy \(c=Uc'\) and hence \(c'=U^\dagger c\). Substitute into the norm:
An operator matrix transforms as \(A'=U^\dagger AU\). Multiplication order matters. A Hermitian matrix satisfies \(A^\dagger=A\); its eigenvalues are real and eigenvectors of distinct eigenvalues are orthogonal. This is the finite-dimensional model for observables. Degenerate eigenvectors need an orthonormal choice within their shared eigenspace; an eigensolver's arbitrary rotation there does not create a different physical energy.
3. A worked interference calculation
Take \(|\psi\rangle=(|0\rangle+e^{i\theta}|1\rangle)/\sqrt2\). In the original basis both probabilities equal \(1/2\). Now measure in \(|\pm\rangle=(|0\rangle\pm|1\rangle)/\sqrt2\). Projection gives
At zero phase the plus outcome is certain; at phase \(\pi\) it is impossible. At phase \(\pi/2\) both outcomes have equal probability. The calculation uses no trajectory or measurement apparatus model; it tests consistent amplitude bookkeeping. A classical mixture with equal populations has \(p_+=1/2\) for every phase because it has no coherence to rotate.
4. Scale before solving
Introduce \(x=L\xi\) and \(E_L=\hbar^2/(2mL^2)\). Then a one-dimensional Hamiltonian becomes \(H/E_L=-d^2/d\xi^2+V(L\xi)/E_L\). The derivative acquires \(1/L\) twice. A dimensionless answer must be multiplied by \(E_L\) to recover energy. Atomic units set \(\hbar=m_e=e=4\pi\epsilon_0=1\); they simplify algebra rather than abolish dimensions. Record the unit system before comparing energies or forces from different software.
5. Exercises with reasoning
Normalize \((1,i,2)^T\) and find the third-component probability. The norm squared is \(1+1+4=6\), so divide by \(\sqrt6\); the requested probability is \(4/6=2/3\). Complex conjugation is essential: \(i^*i=1\), not \(-1\).
A wavefunction has dimension \(L^{-1/2}\) in one dimension. Why? Its squared modulus times \(dx\) must be dimensionless. In three Cartesian dimensions the corresponding dimension is \(L^{-3/2}\). This is a density dimension, not an energy dimension.
6. Misconceptions and next steps
Orthogonality is a relation between states, not a claim that spatial plots never overlap. Two overlapping oscillatory functions can integrate to zero. A basis expansion need not be an approximation when the basis is complete; truncation creates the approximation. Infinite-dimensional operators require domains and boundary conditions in addition to formal matrix analogies. The next lesson makes these mathematical objects into probabilities and observables.
Further conceptual check
An overlap matrix is needed when basis functions are not orthogonal: the norm is \(c^\dagger Sc\), not \(c^\dagger c\). For independent functions \(S\) is positive definite. A zero overlap eigenvalue identifies a redundant direction rather than a new physical state. Keep this distinction when moving from abstract quantum vectors to atomic-orbital coefficients.
7. References and study connections
The derivations and toy arithmetic are original teaching constructions. No molecular simulation is reported here.
8. Related theory and practice
Molecular methods · Quantum Monte Carlo · Gaussian · VASP