Postulates of Quantum Mechanics

1.1 Postulates of Quantum Mechanics

Summary: All of quantum mechanics follows from a small set of assumptions, which cannot themselves be derived.

There is no unique formulation or even number of postulates, but all formulations I’ve seen have the same basic content. This formulation follows Shankar most closely, though he puts III and IV together. Nothing signiﬁcant should be read into my separating them (as many other authors do), it just seems easier to explore the consequences bit by bit.

I: The state of a particle is given by a vector $| ψ (t) ⟩$ in a Hilbert space. The state is normalised: $⟨ ψ (t) | ψ (t) ⟩ = 1$ .

This is as opposed to the classical case where the position and momentum can be speciﬁed at any given time.

This is a pretty abstract statement, but more informally we can say that the wave function $ψ (x, t)$ contains all possible information about the particle. How we extract that information is the subject of subsequent postulates.

The really major consequence we get from this postulate is superposition, which is behind most quantum weirdness such as the two-slit experiment.

II: There is a Hermitian operator corresponding to each observable property of the particle. Those corresponding to position $\hat{x}$ and momentum $\hat{p}$ satisfy $[{\hat{x}}_{i}, {\hat{p}}_{j}] = i ℏ δ_{i j}$ .

Other examples of observable properties are energy and angular momentum. The choice of these operators may be guided by classical physics (eg $\hat{p} \cdot \hat{p} ∕ 2 m$ for kinetic energy and $\hat{x} \times \hat{p}$ for orbital angular momentum), but ultimately is veriﬁed by experiment (eg Pauli matrices for spin- $\frac{1}{2}$ particles).

The commutation relation for $\hat{x}$ and $\hat{p}$ is a formal expression of Heisenberg’s uncertainty principle.

III: Measurement of the observable associated with the operator $\hat{Ω}$ will result in one of the eigenvalues $ω_{i}$ of $\hat{Ω}$ . Immediately after the measurement the particle will be in the corresponding eigenstate $| ω_{i} ⟩$ .

This postulate ensures reproducibility of measurements. If the particle was not initially in the state $| ω_{i} ⟩$ the result of the measurement was not predictable in advance, but for the result of a measurement to be meaningful the result of a subsequent measurement must be predictable. (“Immediately” reﬂects the fact that subsequent time evolution of the system will change the value of $ω$ unless it is a constant of the motion.)

IV: The probability of obtaining the result $ω_{i}$ in the above measurement (at time $t_{0}$ ) is $| ⟨ ω_{i} | ψ (t_{0}) ⟩ |^{2}$ .

If a particle (or an ensemble of particles) is repeatedly prepared in the same initial state $| ψ (t_{0}) ⟩$ and the measurement is performed, the result each time will in general be diﬀerent (assuming this state is not an eigenstate of $\hat{Ω}$ ; if it is the result will be the corresponding $ω_{i}$ each time). Only the distribution of results can be predicted. The postulate expressed this way has the same content as saying that the average value of $ω$ is given by $⟨ ψ (t_{0}) | \hat{Ω} | ψ (t_{0}) ⟩$ . (Note the distinction between repeated measurements on freshly-prepared particles, and repeated measurements on the same particle which will give the same $ω_{i}$ each subsequent time.)

V: The time evolution of the state $| ψ (t) ⟩$ is given by $i ℏ \frac{d}{d t} | ψ (t) ⟩ = \hat{H} | ψ (t) ⟩$ , where $\hat{H}$ is the operator corresponding to the classical Hamiltonian.

In most cases the Hamiltonian is just the energy and is expressed as $\hat{p} \cdot \hat{p} ∕ 2 m + V (\hat{x})$ . (They diﬀer in some cases though - see texts on classical mechanics such as Kibble and Berkshire.) In the presence of non-conservative forces such as magnetism the Hamiltonian is still equal to the energy, but its expression in terms of $\hat{p}$ is more complicated.

VI: The Hilbert space for a system of two or more particles is a product space.

This is true whether the particles interact or not, ie if the states $| ϕ_{i} ⟩$ span the space for one particle, the states $| ϕ_{i} ⟩ \otimes | ϕ_{j} ⟩$ will span the space for two particles. If they do interact though, the eigenstates of the Hamiltonian will not just be simple products of that form, but will be linear superpositions of such states.

References

Shankar 4

[next] [up] [top]