Quantum State Space

Dirac Notation

Comment

Dirac notation, is a notation for linear algebra and linear operators on complex vector spaces together with their dual space both in the finite-dimensional and infinite-dimensional case. In quantum mechanics, Dirac notation is used ubiquitously to denote quantum states.

Bra and Ket

A ket is of the form , it denotes a vector in a complex Hilbert space , and physically it represents a state of some quantum system. A bra is of the form , it denotes a linear map . Letting the linear functional act on a vector is written as .

Quantum State

A quantum state is a quantum wavefunction , represented by a ket . The inner product is then written as . And the expectation value of some observable . The Schrödinger equation becomes

Hilbert Space as Quantum State Space

Quantum State Space

A quantum state space is a complex separable Hilbert space and eigenstates as axes that the state of system is represented by a column vector in this Hilbert space: And the bra is the Hermitian adjoint: . Thus it follows that Any operator then can be represented as a matrix:

e.g. For one degree of freedom this space can be taken to be the space of wave functions (complex-valued functions of a position variable ) in .

Proposition

The th entry of an operator is given by

Proof Can be easily shown by matrix multiplication.

Proposition

An operator is diagonal in the representation of its observable.

e.g. The energy operator is diagonal in the energy representation.

Proposition

The identity operator can be written as .

Unitary Operator

A unitary operator is an operator that satisfies: That is its inverse is equal to its Hermitian adjoint.

Proposition

A unitary operator preserves the inner product. Thus it does not change the length of a vector it operates on.

Proof Consider the unitary operator and two vectors and . Then we have Therefore,

Theorem

Schrödinger’s equation ensures unitary evolution.

Proof Schrödinger’s equation says you start with some state vector at , this state vector then evolves by some complicated evolution, but the length of the vector, which is the probability density is conserved.

Hermitian Operators and Observables

Hermitian Operator

An operator is said to be Hermitian if it satisfies

Proposition

The eigenvalues of Hermitian operators are real.

Proof Suppose is a normalized eigenfunction of associated with eigenvalue , then $$\begin{aligned} && \langle\psi_{n}|\hat{\Theta}|\psi_{n}\rangle &=\left(\langle\psi_{n}|\hat{\Theta}|\psi_{n}\rangle\right)^{} \ \implies && \lambda\langle\psi_{n}|\psi_{n}\rangle &= \lambda^{} \langle\psi_{n}|\psi_{n}\rangle \ \implies&&\lambda &= \lambda^{*} \end{aligned}$$$\square$

Lemma

An operator is Hermitian iff the Hermitian adjoint of it is itself. i.e. .

Theorem

In quantum mechanics, all operators which represent observables are both linear and hermitian.

Proposition

The eigenfunctions of a Hermitian operator corresponding to different eigenvalues are orthogonal.

Proof