Introduction: The Generator of Time
How do quantum states change with time? The answer involves the Hamiltonian operator \(\hat{H}\)—the quantum version of total energy. Just as momentum generates translations in space, energy generates translations in time.
The Schrödinger Equation
The time evolution of a quantum state is governed by:
\[i\hbar\frac{\partial}{\partial t}|\psi(t)\rangle = \hat{H}|\psi(t)\rangle\]This is the time-dependent Schrödinger equation—the equation of motion for quantum mechanics.
The Time Evolution Operator
For time-independent \(\hat{H}\), the formal solution is:
\[|\psi(t)\rangle = \hat{U}(t)|\psi(0)\rangle\]where the time evolution operator is:
\[\hat{U}(t) = e^{-i\hat{H}t/\hbar}\]Worked Examples
Example 1: Energy Eigenstate Evolution
If \(\hat{H}|E_n\rangle = E_n|E_n\rangle\), then:
\[|E_n(t)\rangle = e^{-iE_nt/\hbar}|E_n\rangle\]Energy eigenstates just pick up a phase—they're stationary states!
Example 2: Superposition Evolution
If \(|\psi(0)\rangle = c_1|E_1\rangle + c_2|E_2\rangle\):
\[|\psi(t)\rangle = c_1 e^{-iE_1 t/\hbar}|E_1\rangle + c_2 e^{-iE_2 t/\hbar}|E_2\rangle\]The relative phase changes at frequency \((E_2 - E_1)/\hbar\), causing quantum beats.
Example 3: Probability Conservation
Since \(\hat{U}\) is unitary:
\[\langle\psi(t)|\psi(t)\rangle = \langle\psi(0)|\hat{U}^\dagger\hat{U}|\psi(0)\rangle = \langle\psi(0)|\psi(0)\rangle = 1\]Probability is conserved under time evolution.
The Quantum Connection
The Hamiltonian is the central operator in quantum mechanics because it determines time evolution. Finding the Hamiltonian and its eigenstates is the primary task in solving any quantum problem. Energy eigenstates are special: they don't change with time (except for an unmeasurable phase), which is why they're called stationary states.