An operator is a transformation that takes a function as an input and produces another function (usually).

Transcription

1 Formalism of Quantum Mechanics Operators Engel 3.2 An operator is a transformation that takes a function as an input and produces another function (usually). Example: In QM, most operators are linear: a, b constants Multiplication by x: The hat ^ is usually there to show that it is an operator. For example the total energy is called a Hamiltonian, H, is one formulation of CM by Hamilton. In QM H the Hamiltonian for the total energy is an operator. Very important (later)

2 Order matters! Also Example: In QM an observable; i.e., any quantity that can be measured has an associated linear operator. (See Table 3.1 Engel; p. 69).

3 Commutators Engel 6.1 p. 119 Given 2 operators: AA and BB The commutator (which is also an operator) is defined as: If one says the operators commute. Then one can measure the properties associated with AA and BB precisely at the same time If one says the operators do not commute. Then one cannot measure the properties associated with AA and BB precisely at the same time This is related to the Heisenberg Uncertainty Principle (more later)

4 Eigenvalues and eigenfunctions Engel 2.4; p. 55) In general, an operator operating on a function gives a different function In certain cases, the operator operating on a function gives the same function multiplied by a constant. That function is an eigenfunction of the operator, and the constant is an eigenvalue. These are important in measurements. = an eigenvalue equation Example: What about e ax2? No!

5 Example: The eigenvalue (-a 2 ) has more than one eigenfunction. This is related to the concept of degeneracy (later). To solve an eigenvalue equation multiply by f * and integrate: if f is normalized (later)

6 Example: Show that is an eigenfunction of and find its eigenvalue The function is an eigenfunction if it obeys: Eigenvalue = -1

7 Hermitian Operators Engel 3.2; p. 69 Hermitian operators are those operators with real eigenvalues. In QM they represent observables because the outcome of a measurement is real Definition: An operator is Hermitian if Example: is d/dx Hermitian for eigenfunction: e ikx? No because the eigenvalue is imaginary.

8 Show that all the eigenvalues of a Hermitian operator are real. If A is Hermitian Only possible if a = a *

9 Example: Consider the fact that all wavefunctions in QM are square integrable meaning they go to 0 at the limits of their defined intervals Is a Hermitian operator for functions on the x-axis which go to 0 at infinity? Write down the definition of Hermitian for the operator:? LHS Integration by parts =RHS

10 It can deduced therefore that px is Hermitian for the space of all functions that are square integrable. Orthogonality Engel 2.5; p. 57 In a 3D vector space 2 vectors are orthogonal if the angle between them is 90 o (also called perpendicular) Their dot product Two functions are orthogonal if For a Hermitian operators there eigenfunctions ø i (x), i = 1, 2, such that

11 Example: Show that the set of functions: is orthogonal if m and n are integers Show that a) If m = n b) If m n So they are orthogonal. See problem 2.6 in Engel.

12 Show that the eigenfunctions corresponding to a Hermitian operator are orthogonal Let Ω be an operator with eigenfunctions f, g with eigenvalues ω 1 and ω 2 But Hermitian means that: Real eigenvalues for Hermitian operators

13 But f(x) and g(x) are orthogonal Normalization Engel 2.5; p. 58 A function is said to be normalized if To normalize a function is to scale it so that it is normalized. This comes from the Born statistical interpretation of the wavefunction. If a set of functions is orthogonal and each function is normalized the set is said to be orthonormal Kronecker delta function

14 Example: Normalize the function on the interval (0, ) Example: Normalize the function over the interval 0 x a

15 The need to normalize wave functions will become apparent when we discuss the meaning of wave functions in QM: The square of the wave function is a measure of the probability of finding the particle in a certain location. Since the particle has to be somewhere the integral of the probability over the entire space must be 1 (i.e. 100%).