The Hamiltonian and Time Evolution in Quantum Mechanics

The Hamiltonian and Time Evolution in Quantum Mechanics#

In quantum mechanics, the state of a physical system — such as a particle or an atom — is described by a mathematical object called a state vector, usually written as \(\ket{\psi(t)}\). This state can change or evolve over time, and the rules for how it changes are given by a fundamental equation in quantum mechanics called the time-dependent Schrödinger equation:

\[ i\hbar\frac{d}{dt}\ket{\psi(t)} = H(t)\ket{\psi(t)} \]

Let’s break this down:

  • \(\hbar\) is the reduced Planck’s constant, which is Planck’s constant \(h\) divided by \(2\pi\). It appears throughout quantum mechanics and ensures the equation has correct units.

  • \(\ket{\psi(t)}\) is the state of the system at time \(t\). It’s a vector in a complex vector space called Hilbert space.

  • The operator \(H(t)\), called the Hamiltonian, plays a central role. It represents the total energy of the system — both kinetic (motion) and potential (position-dependent forces).

  • The left-hand side of the equation is the rate of change of the state with respect to time, and the right-hand side tells us how the Hamiltonian causes this change.

In essence, the Schrödinger equation is the quantum version of Newton’s laws — it tells us how a system evolves from one moment to the next.

What is the Hamiltonian?#

The Hamiltonian is a special kind of operator: it is Hermitian, which means it obeys the mathematical condition \(H^\dagger=H\). This ensures that it has real eigenvalues and a complete set of orthonormal eigenstates. This is important because it means the outcomes of energy measurements (the eigenvalues) are always real numbers — just as we observe in experiments.

If you’ve studied classical physics, you might recognize the Hamiltonian from classical mechanics, where it’s a function representing the total energy of a system in terms of coordinates and momenta. In quantum mechanics, the Hamiltonian plays a similar role, but it’s an operator rather than a function.

Time Evolution When the Hamiltonian is Constant#

If the Hamiltonian does not change with time — such as when a particle (e.g. electron) moves in a fixed external field (e.g. the field produced by a proton) — we say it is time-independent, and the evolution of the state has a particularly elegant form.

Let’s say the Hamiltonian \(H\) has a set of eigenstates \(\ket{E_n}\), each with a corresponding energy eigenvalue \(E_n\). These eigenstates satisfy:

\[ H\ket{E_n} = E_n\ket{E_n} \]

Any state \(\ket{\psi(0)}\) can be written as a superposition (a linear combination) of these eigenstates:

\[ \ket{\psi(0)} = \sum_n \braket{E_n|\psi(0)} \ket{E_n} \]

Then, at any later time \(t\), the state evolves as:

\[ \ket{\psi(t)} = \sum_n e^{-iE_n t/\hbar} \braket{E_n|\psi(0)} \ket{E_n} \]

Here’s what’s happening:

  • Each energy component of the state simply picks up a phase factor \(e^{-iE_n t/\hbar}\) over time.

  • The probability of measuring a particular energy doesn’t change — only the relative phases between components do.

  • These phase changes lead to interference effects, which are responsible for much of the rich behavior in quantum systems.

Summary#

  • The Schrödinger equation describes how quantum states evolve over time.

  • The Hamiltonian is the operator associated with the system’s energy and drives the time evolution.

  • When the Hamiltonian is time-independent, the solution is a simple sum over energy eigenstates with time-dependent phase factors.

  • This framework underpins many areas of quantum physics, from atomic structure to quantum computing.

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