Semester 1: Postulates

Question 1

Define TQW Postulate 1 (quantum state)

Answer 1

The state of a non-relativistic quantum mechanical particle at time, t, is specified by a continuous, single-valued, complex function ψ(r, t). This function, called the wavefunction or state function, has the property that ψ*(r, t)ψ(r, t)dτ is the probability that the particle lies in the volume element dτ located at r at time t.

Question 2

Define TQW postulate 2 (observables)

Answer 2

To every physically measurable quantity, o, called an observable or dynamical variable, there corresponds a linear Hermitian operator, Ô, whose eigenfunctions form a complete basis and whose eigenvalues, by virtue of the Hermiticity of the operator, are real-valued.

Question 3

Define TQW postulate 3 (TDSE)

Answer 3

The state of the system evolved in time according to the time-dependent Schrödinger equation. The eigenfunctions of the Hamiltonian operator are stationary states

Ĥ = Hamiltonian operator
uₙ = eigenfunctions (which can be found by solving the TISE: Ĥuₙ = Euₙ)

Question 4

Define TQW postulate 4 (eigenvalue probabilities)

Answer 4

When a measurement of an observable represented by the Hermitian operator, Ô, is carried out on a system whose wavefunction is ψ=∑cₙuₙ, the probability of the result being equal to a discrete eigenvalue, oₙ, is |cₙ|².

uₙ = eigenfunctions
oₙ = eigenvalues

If the system is in an eigenstate of the operator, the eigenvalue associated with that eigenstate is measured with complete certainty.

By contrast, for a continuous eigenvalue spectrum, where ψ(x) = ∫F(k)φ(k, x)dk, the probability of a result in the range between o(k) and o(k + dk) is |F(k)|² dk.

Question 5

Define TQW postulate 5 (Born rule)

Answer 5

If the result of a measurement is oₙ (i.e. an eigenvalue of the operator Ô) with an associated eigenfunction χₙ, the state of the system immediately after the measurement becomes χₙ. (The wavefunction of the system collapses to the eigenfunction χₙ). The probability of finding the system at time, t, in the state χₙ is given by the modulus squared of the value of the overlap integral: |∫ χₙ*ψ(x, t)dx|². This is the Born rule.