Mathematical Formulation of Quantum Mechanics

The Wave Function

In quantum mechanics, a physical system is described by a wave function, typically denoted as Ψ (Psi). The wave function is a complex-valued function of space and time that contains all the information about the quantum system.

The physical interpretation of the wave function comes from Born's rule, which states that the squared magnitude of the wave function gives the probability density of finding the particle at a particular position.

The probability of finding a particle in a small volume dV around position r:

P(r) dV=∣Ψ(r,t)∣2 dVP(\mathbf{r})\,dV = |\Psi(\mathbf{r},t)|^2\,dV

For a wave function to be physically meaningful, it must be:

  • Single-valued (one probability value per position)
  • Continuous and smooth (except at points of infinite potential)
  • Normalizable (total probability equals 1)

Normalization condition:

∫−∞∞∣Ψ(r,t)∣2 dV=1\int_{-\infty}^{\infty} |\Psi(\mathbf{r},t)|^2\,dV = 1

Hilbert Space

Quantum mechanical systems are mathematically described in a complex vector space called Hilbert space. Each quantum state corresponds to a vector in this space.

Key properties of Hilbert space include:

  • It's a complete inner product space
  • State vectors are typically denoted using Dirac's bra-ket notation: |Ψ⟩
  • The inner product ⟨Φ|Ψ⟩ represents the overlap between two quantum states

Inner product in position representation:

⟨Φ∣Ψ⟩=∫−∞∞Φ∗(r)Ψ(r) dV\langle \Phi | \Psi \rangle = \int_{-\infty}^{\infty} \Phi^*(\mathbf{r}) \Psi(\mathbf{r})\,dV

Quantum Operators

Physical observables in quantum mechanics are represented by Hermitian operators acting on the wave function. An operator  transforms one state vector into another.

Common quantum mechanical operators include:

Position operator:

x^Ψ(x)=xΨ(x)\hat{x} \Psi(x) = x \Psi(x)

Momentum operator:

p^=−iℏddx\hat{p} = -i\hbar \frac{d}{dx}

Hamiltonian operator (energy):

H^=−ℏ22m∇2+V(r)\hat{H} = -\frac{\hbar^2}{2m}\nabla^2 + V(\mathbf{r})

The expected value of an observable A in a state |Ψ⟩ is given by:

⟨A⟩=⟨Ψ∣A^∣Ψ⟩\langle A \rangle = \langle \Psi | \hat{A} | \Psi \rangle

The Schrödinger Equation

The time evolution of a quantum system is governed by the Schrödinger equation, which comes in two main forms:

Time-dependent Schrödinger equation:

iℏ∂∂tΨ(r,t)=H^Ψ(r,t)i\hbar \frac{\partial}{\partial t}\Psi(\mathbf{r},t) = \hat{H}\Psi(\mathbf{r},t)

Time-independent Schrödinger equation (for stationary states):

H^Ψ(r)=EΨ(r)\hat{H}\Psi(\mathbf{r}) = E\Psi(\mathbf{r})

Where ℏ is the reduced Planck constant, Ĥ is the Hamiltonian operator, and E is the energy.

Eigenvalues and Eigenstates

When a quantum system is measured, the wave function collapses to an eigenstate of the measured observable, and the measurement result is the corresponding eigenvalue.

Eigenvalue equation:

A^∣an⟩=an∣an⟩\hat{A}|a_n\rangle = a_n|a_n\rangle

Where |an⟩ is an eigenstate of the operator  with eigenvalue an.

The probability of measuring the eigenvalue an when the system is in state |Ψ⟩ is:

P(an)=∣⟨an∣Ψ⟩∣2P(a_n) = |\langle a_n | \Psi \rangle|^2

Uncertainty Relations

For any two non-commuting operators  and B̂, the following uncertainty relation holds:

ΔAΔB≥12∣⟨[A^,B^]⟩∣\Delta A \Delta B \geq \frac{1}{2}|\langle[\hat{A}, \hat{B}]\rangle|

Where ΔA and ΔB are the standard deviations of measurements, and [Â,B̂] is the commutator.

The most famous uncertainty relation is Heisenberg's uncertainty principle for position and momentum:

ΔxΔpx≥ℏ2\Delta x \Delta p_x \geq \frac{\hbar}{2}

Matrix Mechanics

An alternative but equivalent formulation of quantum mechanics is matrix mechanics, developed by Heisenberg, Born, and Jordan. In this formulation, physical observables are represented by matrices.

When working with a discrete basis, quantum mechanical operators can be represented as matrices:

Amn=⟨m∣A^∣n⟩A_{mn} = \langle m | \hat{A} | n \rangle

The evolution of the system is described by the Heisenberg equation of motion:

dA^dt=iℏ[H^,A^]+∂A^∂t\frac{d\hat{A}}{dt} = \frac{i}{\hbar}[\hat{H}, \hat{A}] + \frac{\partial \hat{A}}{\partial t}