Math Infinitum

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Intuition

Between measurements a quantum state changes smoothly and deterministically. The rule is one equation: the rate of change of the state is the energy operator acting on it, divided by $i\hbar$ . Know the state now and the equation fixes it at every later time, and because the energy operator is Hermitian the total probability never changes.

A wind map says which way the air moves at every point; drop a feather and its whole path follows. The Schrödinger equation is the wind map of state space, and the Hamiltonian is the wind.

The law of motion

The state of a closed system obeys the time-dependent Schrödinger equation, in which $\hat{H}$ , the Hamiltonian, is the observable of energy. It is linear, so superpositions of solutions are solutions, and first order in time, so the state at one moment determines it at every other.

i\hbar\,\frac{d}{dt}\ket{\psi(t)}=\hat{H}\ket{\psi(t)}

What the equation says

It holds between measurements. A measurement changes the state by the rule of the observables chapter, which is not of this form.
Linearity: if $\ket{\psi_{1}(t)}$ and $\ket{\psi_{2}(t)}$ are solutions, so is .

Total probability is conserved

\frac{d}{dt}\braket{\psi(t)|\psi(t)}=0

Differentiate the squared norm with the product rule. The Schrödinger equation gives the derivative of the ket and, by taking the adjoint, of the bra; the two terms are then the same matrix element of the Hamiltonian with opposite signs, because the Hamiltonian is Hermitian.

Proof steps

\frac{d}{dt}\braket{\psi|\psi}=\Bigl(\frac{d}{dt}\bra{\psi}\Bigr)\ket{\psi}+\bra{\psi}\Bigl(\frac{d}{dt}\ket{\psi}\Bigr)

The product rule for a bracket.

\frac{d}{dt}\ket{\psi}=\frac{1}{i\hbar}\hat{H}\ket{\psi}

The Schrödinger equation.

\frac{d}{dt}\bra{\psi}=-\frac{1}{i\hbar}\bra{\psi}\hat{H}

Its adjoint: $\hat{H}^{\dagger}=\hat{H}$ and $(1/i)^{*}=-1/i$ .

\frac{d}{dt}\braket{\psi|\psi}=-\frac{1}{i\hbar}\bra{\psi}\hat{H}\ket{\psi}+\frac{1}{i\hbar}\bra{\psi}\hat{H}\ket{\psi}=0

The two terms cancel.

Applications

Practice

One Equation

The rate of change of a quantum state is the Hamiltonian acting on it, divided by $i\hbar$ . The Hamiltonian is the observable of energy.

i\hbar\,\frac{d}{dt}\ket{\psi}=\hat{H}\ket{\psi}

Try it

Which equation governs a closed quantum system between measurements?