Formalism of Quantum Mechanics

Transcription

1 Dirac Notation Formalism of Quantum Mechanics We can use a shorthand notation for the normalization integral I = "! (r,t) 2 dr = "! * (r,t)! (r,t) dr =!! The state! is called a ket. The complex conjugate of the ket is called a bra More generally, the scalar product of two states is written as! 1! 2 = "! * 1 (r,t)! 2 (r,t) dr = "! * 1 (p,t)! 2 (p,t) dp! Note that the Dirac notation is independent of the representation of the states. However, for calculation purposes, one must pick a particular representation. Prove that! 1! 2 =! 2! 1 *

2 Hilbert space Formalism of Quantum Mechanics If!! = N Where N is a finite real number, then we say that the function ψ is square-integrable The vector space which includes all square-integrable functions is called Hilbert space Wave functions must be square integrable in order to be normalizable. Therefore wave functions live in Hilbert space

3 Dynamical variables and linear operators Every dynamical variable is associated with a linear operator which can operate on wave functions. Examples: In the position representation, Position : Momentum :! i! " "x Examples: In the momentum representation, x Position : i!!!p Momentum : When an operator A acts on a state (wave function) ψ, it results in a new wave function χ: p A! = " A(c 1! 1 + c 2! 2 ) = c 1 ( A! 1 ) + c 2 ( A! 2 ) (linearity)

4 Dynamical variables and linear operators (position representation)

5 Dynamical variables and linear operators Certain wave functions ψ n are said to be eigenfunctions of A if A acting on ψ n, results in the same wave function ψ n multiplied by a constant a n A! n = a n! n The constants a n are the eigenvalues of A. The set of all eigenvalues is called the spectrum of A. Example: Time-independent Schrodinger equation: The total energy operator H acting on energy eigenfunctions results in the same eigenfunctions multiplied by energy eigenvalues: H! n = E n! n

6 Expectation value of a variable The average or expectation value of measurements of a dynamical observable associated with the operator A on an ensemble of identical systems in the state ψ is compactly written in Dirac notation as A = "! * (x)a! (x)dx =! A!

7 Measurement of dynamical variables The outcome of a measurement of a dynamical variable represented by the operator A can only be one of the eigenvalues of A. The spectrum of the observable can be discrete or continuous. Only real values can be measured. Hence operators corresponding to observable variables have a real spectrum. Operators with real eigenvalues corresponding to physically observable variables are Hermitian

8 Hermitian Operator A Hermitian operator A has real eigenvalues. A series of measurements of A will have real outcomes. The expectation (average) value of a Hermitian operator is thus real. A = A * A =! A! = " A * =! A! * =! * (x)(a! (x))dx " " (A! (x)) *! (x)dx "! * (x)(a! (x))dx = (A! (x)) *! (x)dx! A! = A!! In general, Hermitian operators are defined by the condition #! * (r)(a" (r))dr = # (A!(r)) * " (r)dr! A" = A! "

9 Adjoint operator Formalism of Quantum Mechanics An adjoint operator A For Hermitian operators, (A-dagger) is defined by the condition #! * (r)(a" (r))dr = # (A!(r)) * " (r)dr #! * (r)(a" (r))dr = # (A!(r)) * " (r)dr A = A self-adjoint This condition can be used to check whether an operator is Hermitian. Example: Check that the momentum operator is Hermitian

10 Unitary operator A unitary operator U has the property that U = U!1 The operator U -1 is the inverse defined by U!1 U = UU!1 = I The operator I is the identity operator defined by I! =! Thus U U! = UU! = I! =! A unitary operator can be written in terms of a Hermitian operator as U = e ia We will see that the operator for evolving a wave function over time is unitary and time-reversible.

11 Projection operator A projection operator P is a Hermitian operator that has the additional property that it is equal to its square. P 2 = P If P is a projection operator, then I-P is also a projection operator. Check. We will discuss projection operators in more detail later.