Physics 3 for Electrical Engineering Ben Gurion University of the Negev Lecturers: Daniel Rohrlich, Ron Folman Teaching Assistants: Daniel Ariad, Barukh Dolgin Week 8. Quantum mechanics – raising and lowering operators, 1D harmonic oscillator harmonic oscillator eigenvalues and eigenfunctions matrix representation motion of a minimum- uncertainty wave packet 3D harmonic oscillator classical limit Sources: Merzbacher (2 nd edition) Chap. 5 Sects. 1-4; Merzbacher (3 rd edition) Chap. 5 Sects. 1, 3 and Chap. 10 Sect. 6; Tipler and Llewellyn, Chap. 6 Sect. 5.

Let’s see and compare how these two different methods apply to the quantum harmonic oscillator. We already stated that Schrödinger’s wave equation, with its continuous solutions, and Heisenberg’s matrix algebra, with its quantum jumps, are equivalent.

The 1D harmonic oscillator x 0

x 0 x 0 x 0 x 0 x 0 x 0 t1t1 t3t3 t4t4 t5t5 t6t6 t2t2

x 0 x 0 x 0 x 0 x 0 F(t6)F(t6) F(t4)F(t4) F(t2)F(t2) F(t3)F(t3) x 0 t1t1 t3t3 t4t4 t5t5 t6t6 t2t2

This system is a model for many systems, e.g. molecules made of two atoms. x 0

x 0 The 1D harmonic oscillator Any system with a potential minimum at some x = x 0 may behave like a harmonic oscillator at low energies: V(x) = V(x 0 ) + (x–x 0 ) 2 V′′(x 0 ) + …. x0x0

x 0 The 1D harmonic oscillator Also, a mode of the electromagnetic field of frequency ν behaves like a 1D harmonic oscillator of frequency ν. Its energy levels nhν correspond to n photons of frequency ν.

The 1D harmonic oscillator Schrödinger’s equation: x 0

Solving Schrödinger’s equation Schrödinger’s way: Define a new variable, where In terms of ξ, Schrödinger’s equation is Solutions: Try where H(ξ) is a polynomial. ψ is a solution when H(ξ) is one of the Hermite polynomials: H 0 (ξ) = 1, H 1 (ξ) = 2ξ, H 2 (ξ) = 4ξ 2 –2, H 3 (ξ) = 8ξ 3 –12ξ, ….

Solving Schrödinger’s equation Heisenberg’s way: Define new variables where Let’s prove that Try also to prove (for any three operators ) that

Raising and lowering operators To prove:

Raising and lowering operators To prove:

Raising and lowering operators To prove:

Suppose is an eigenstate of with eigenvalue E. Then using Therefore, is an eigenstate of with eigenvalue. We call a raising operator. Homework: Show that is a lowering operator, i.e.

Suppose is an eigenstate of with eigenvalue E. Then using Therefore, is an eigenstate of with eigenvalue. We call a raising operator. The harmonic oscillator must have a ground state – call it ψ 0 (x) or – with minimum energy. For we have which means and therefore

Since and, the ground-state energy is, and the energy of the state, defined by is From we learn that Thus we call the number operator. Since we can write. Likewise, since, we can write

Since and, the ground-state energy is, and the energy of the state, defined by is Normalization: So the ground state normalization is Then for all n, (Prove it!)

What are the lowest eigenstates of the harmonic oscillator? Note that the eigenfunctions ψ n (x) are even or odd in x. Why? Harmonic oscillator eigenvalues and eigenfunctions

This Figure is taken from here. here Harmonic oscillator eigenvalues and eigenfunctions

This Figure is taken from here. here If the harmonic oscillator represents a mode of the electromagnetic field, an energy level ½ represents n photons each having energy, plus additional “zero-point energy” of per mode. Harmonic oscillator eigenvalues and eigenfunctions

Matrix representation In the basis of harmonic-oscillator eigenvectors, we can represent the operators as matrices. Since ½ if m = n and vanishes otherwise, we can represent as an infinite matrix:

Matrix representation In the basis of harmonic-oscillator eigenvectors, we can represent the operators as matrices. Since if m = n–1 and vanishes otherwise, we can represent as an infinite matrix:

Matrix representation In the basis of harmonic-oscillator eigenvectors, we can represent the operators as matrices. Since if m = n+1 and vanishes otherwise, we can represent as an infinite matrix:

The normalized harmonic-oscillator eigenvectors themselves are the basis vectors:

The transpose of a matrix is written and defined by

The adjoint of a matrix is written and defined by

The adjoint of a matrix is written and defined by Any observable is self-adjoint, i.e. “ is Hermitian” and “ is self-adjoint” mean the same thing.

The adjoint of a matrix is written and defined by Any observable is self-adjoint, i.e. “ is Hermitian” and “ is self-adjoint” mean the same thing. The raising and lowering operators and are not self- adjoint, but are adjoints of each other:

are manifestly self-adjoint:

Vectors, too, have adjoints. For any we have and, for example, Try to prove, for any two operators, the rule

Motion of a minimum-uncertainty wave packet The ground state wave function ψ 0 (x) is a minimum-uncertainty wave function: we can calculate In general, the time evolution of any initial wave function Ψ(x,0) can be obtained from the expansion of Ψ(x,0) in the basis of energy eigenstates. If the initial wave function is for given c n, then the wave function at time t is

Motion of a minimum-uncertainty wave packet The period of a classical harmonic oscillator having angular frequency ω is T = 2π /ω. If we add T to t in Ψ(x,t), the wave function does not change, up to an overall phase factor –1: Schrödinger even found a solution Ψ(x,0) that moves between x = R and x = –R while the probability distribution |Ψ(x,0)| 2 keeps its (minimum uncertainty) shape:

Motion of a minimum-uncertainty wave packet Schrödinger thought his solution might be typical, but it is not. Usually the probability distribution spreads over time. Prove that a free 1D wave packet spreads, that if it is initially then at time t the wave packet is Show that Δp = is constant in time, but

3D harmonic oscillator The Schrödinger equation for a general 3D harmonic oscillator, allows for a harmonic potential with different strengths along the three axes. Show that the eigenfunctions are products of 1D harmonic oscillator eigenfunctions. What are the lowest energies and their degeneracies as a function of k x, k y and k z ?

Classical limit A classical particle in a square well has equal probability to be at any point inside. How about a quantum particle in its ground state? a quantum particle in a highly excited state? The probability P(x) for classical harmonically oscillating particle to be at any point x is inversely proportional to its speed at that point: P(x) ~ (2E/m – ω 2 x 2 ) –1/2, where E is total energy. n = 0n = 10.