My previous posts on quantum mechanics, and specifically uncertainty principles, were essentially about the quantum state ψ of the system at a fixed time. This post concerns the time evolution of quantum systems.
The key evolution equation here is a Hilbert-space-valued ordinary differential equation that consists of two key ingredients: the differential operator i ℏ ∂t (the energy operator) and a Hamiltonian operator H that describes the energetics of the system. (Sorry, “H” has changed from denoting a Hilbert space to denoting the Hamiltonian operator. So many concepts, so few letters…) The time-dependent Schrödinger equation with Hamiltonian H is
i ℏ ∂tψ = Hψ.
Often, the Hamiltonian is itself a differential operator: a good example is the Schrödinger equation for a single non-relativistic particle of mass m moving in a scalar potential V: ℝn → ℝ:
i ℏ ∂tψ = − (ℏ2 ⁄ 2m) Δψ + V ψ
The Hamiltonian in this case is the familiar “kinetic energy + potential energy” one. V is fairly obviously the potential energy term. The kinetic energy term is the usual “½ × mass × velocity2” but in an interesting form: it is 1⁄2m times the dot product of the momentum operator P := −iℏ∇ with itself, hence the “− (ℏ2 ⁄ 2m) Δ”, where Δ denotes the spatial Laplacian.
Simply put, the time-dependent Schrödinger equation is an absolute pain to solve in all but the simplest settings. Life gets slightly easier if we search for so-called “stationary states”, which, despite the name, are not actually solutions ψ that are constant in time, but rather are eigenstates of the Hamiltonian operator H. (In a sense to be made precise in a moment, these stationary states are constant from the point of view of any observation operator, even though they are themselves non-constant.)
A stationary state is a solution ψ to the time-independent Schrödinger equation, i.e. to the eigenvalue problem
Hψ = Eψ.
Here the eigenvalue E ∈ ℝ is the energy of the quantum state ψ. If H is compact and self-adjoint, then the usual remarks about there being at most countable many eigenvalues apply. In any case, the state ψ with least energy E is called the ground state of the system, and E is called ground state energy or zero-point energy of the system; the other eigenstates are called excited states.
Note that stationary states are not actually constant in time: substituting the definition into the time-dependent Schrödinger equation reveals that a stationary state ψ satisfies the (complex) ordinary differential equation
i ℏ ∂tψ = Eψ,
to which the solution, given ψ at some initial time t0, is
ψ(t) = e−iE(t−t0)⁄ℏ ψ(t0).
So, stationary states actually evolve by “rotation”, with “angular velocity” E⁄ℏ; but note that the probability density |ψ(t)|2 is independent of time t. Indeed, if A is any linear observation operator, then
〈A〉ψ(t) = 〈Aψ(t), ψ(t)〉 = e−iE(t−t0)⁄ℏ e+iE(t−t0)⁄ℏ 〈Aψ(t0), ψ(t0)〉 = 〈A〉ψ(t0).
Example 1: the Free Particle. The time-independent Schrödinger equation for a single particle of mass m moving in (a subset of) ℝ in the constant potential V = 0 (a so-called “free particle”) is the second-order ordinary differential equation
ψ″ = − (2mE ⁄ ℏ2) ψ.
The character of the solution to this ODE depends on the sign of E. If E = 0, then there is no solution; ψ cannot simultaneously be an affine function, as required by for the second derivative to vanish, and have unit norm, as required for normalization. For positive E we get the oscillatory solutions
ψ(x) = C1 ei √(2mE) x ⁄ ℏ + C2 e−i √(2mE) x ⁄ ℏ
with constants of integration C1 and C2 determined by the boundary and normalization conditions. On the other hand, for negative E we get the non-physical exponentially growing solutions
ψ(x) = C1 e√(2m|E|) x ⁄ ℏ + C2 e−√(2m|E|) x ⁄ ℏ
Example 2: the Quantum Harmonic Oscillator. As above, but with the potential V(x) := ½mω2x2. This system has a one-parameter discrete family of solutions, parametrized by n ∈ ℕ0. To keep the lengths of various expressions to a manageable level it helps to nondimensionalize the Schrödinger equation, i.e. work in the natural length and energy scales: if lengths are measured in multiples of (ℏ ⁄ mω)1⁄2 and lengths in multiples of ℏω, then the time-independent Schrödinger equation for the system becomes
−½ψ″ + ½x2ψ = Eψ,
which has a nice family of solutions in terms of the Hermite polynomials Hn(x) := (−1)n ex2 ∂ne−x2: the eigenstates are
ψn(x) = (2n n! √π)−1⁄2 e−x2 Hn(x)
with corresponding eigenvalues (energy levels) En = n+½. In particular, the ground state energy E0 = ½, and the ground state ψ0 has Gaussian probability density function. There’s another funky solution method for the QHO called the “ladder operator method”, due to Paul Dirac, which uses creation and annihilation operators and generalises nicely to other problems: I think that I’ll write a separate post on those later.