Hilbert Space Methods Used in a First

Course in Quantum Mechanics

Victor Polinger Physics/Mathematics

Bellevue College

03/07/13 - 04/11/13

Outline • The Infinite Square Well: A Follow-Up

• Timeline of basic events

• Statistical interpretation of the wave function. Movie.

• Three approaches in Quantum Theory. Basic postulates of Schrődinger’s Wave Mechanics.

• Time-dependent Schrӧdinger Equation in 1D case. Separation of variables.

• Stationary Schrӧdinger Equation and its matrix equivalent. The Hilbert space as the tool of Quantum Theory.

• A Bite of Solid-State Physics. Phonons.

The Problem of Infinite Square Well: A Follow-Up

2

11 1 1

1 1

2 , , 2 2

pha p E

m

Ground state:

1 /

1 1 1

2, , sin

iE t xx t e x x

a a

Excited states:

/

1

2

1

, ,

2sin ,

, 2,3,4,...

niE t

n

n

n

x t e x

nxx

a a

E n E n

De Broglie’s waves: 2

h

p p

In the ground state:

2

1 1 1 1 1

0

2 2 2 2 2 2

1 1 1 1 1 2

0

22

2

1 1 1 1 1 1

0

222 2 2 2

1 1 1 1 1 12

0

22

, ,2

1 1, ,

3 2

1 2,

2 3

, , 0

, ,

,

a

a

a

a

ax x x x x dx

x x x x x dx a

ax x x

dp p p i x x dx

dx

dp p p x x

dx a

p p pa

2 1.136

22 3 2 2

x p

The Uncertainty Principle: 2

x p

Introduction: Timeline of Basic Events - I

• 1858-1859, B. Stewart (England) and G. Kirchhoff (Prussia). Black-body radiation.

• Max Plank (Germany),December 14, 1900, :

• 1905, Lord Rayleigh and J. Jeans (England)

2

5 /

2

1hc kT

c hI

e

346.63 10 J sh

c is speed of light, is the Plank constant.

Plank’s empirical formula:

Light incident into a hole in a cavity is lost forever.

For an observer, the hole looks like a black body

The ultraviolet catastrophe, 4

constI

Electromagnetic energy does not follow the classical description, but can only be emitted in discrete packets of energy, hν, proportional to the frequency, ν.

In Latin, “quantus” means “how much”

Introduction: Timeline of Basic Events - 2

• 1902, P. Lenard (Germany). Photoelectric Effect.

• 1905, A. Einstein (Switzerland)

• 1924, Louis de Broglie (France). Wave-Particle Duality.

Maximum energy of the ejected electrons did not depend on light intensity but did depend on the color -- the higher-frequency light caused electrons to be ejected with more energy (Nobel Prize of 1905). Lenard joined Nazi Party at an early stage. Together with J. Stark he leaded the core campaign to label Einstein’s theory as Jewish Physics

solved this apparent paradox by describing light as composed of discrete quanta, now called photons, rather than continuous waves. Energy carried by one photon is E = hν.

KE

KE

h W

h W

Nobel Prize of 1923

for explanation of the photoeffect

If electromagnetic waves can manifest features of particles, then a moving particle may be associated with a wave (Nobel Prize of 1929). In his own words, “My essential idea was to extend to all particles the coexistence of waves and particles discovered by Einstein in 1905 in the case of light and photons.”

h

p

Nobel Prize of 1929 for the idea of wave-particle duality conceiving the wave

mechanics

Statistical Interpretation of the Wave Function

• 1926, Max Born (Germany). Statistical interpretation (Nobel Prize of 1954)

• 1927, The Davisson-Germer’s Experiment (Bell Labs,USA).

• 1989, Akira Tonomura

Using the surface of the crystal lattice of nickel as diffraction grating, they observed diffraction pattern from a beam of incident electrons

2

,x t dx probability of finding the particle in the interval (x, x + dx) at time t

Nobel prize of 1954 for fundamental research in quantum mechanics, especially for statistical interpretation of the wave function

Nobel prize of 1937 (awarded to Davisson

and Thomson)

for experimental discovery of

electron diffraction Double-slit experiment on electron diffraction

in Hitachi Lab, Japan

Dr. Quantum Explains it All… (watch the movie)

Three Approaches in Quantum Theory.

Matrix Mechanics (1925) M.Born, W. Heisenberg, and P. Jordan in Gőttingen, Germany (The “Drei-Männer Arbeit”).

In 1928 all three authors were nominated for Nobel Prize by A. Einstein. Only Heisenberg

won it in 1932

The Path Integral Formulation (1948) R. Feynman, Princeton,

NJ, USA.

Wave Mechanics (1926) E. Schrődinger, Zűrich, Switzerland.

Their equivalency was proven by Schrődinger

in 1926

Their equivalency was proven by R. Feynman in 1948

W. Heisenberg, Nobel prize of 1932

For creation of Quantum Mechanics

E. Schrődinger Nobel Prize of 1933 (shared with P. A. M. Dirac).

R. Feynman Nobel Prize of 1965 (shared with J. Schwinger and S.-I. Tomonaga)

Basic postulates of Schrődinger’s Wave Mechanics

1. Every mechanical system is completely described by a wave function, Ψ(x, t), representing the state of the system. The wave function is double-integrable over all real numbers; in the case of a finite motion, the integral

with a smooth core, φ(x1, x2), exists (converges). [Ψ(x, t) → 0 as x →±∞]

2. The description of nature is essentially probabilistic. Ψ(x, t) is the amplitude of probability: |Ψ(x, t)|2dx represents the probability of finding the particle in the interval (x, x + dx) at the instant t.

3. For a finite motion, the wave function, Ψ(x, t), is normalized to 1:

4. In the case of a finite and smooth potential energy, V(x), the wave function, Ψ(x, t), and its first derivatives, ∂Ψ(x, t)/∂x and ∂Ψ(x, t)/∂t, is continuous, single-valued, and finite.

Mathematically rigorous formulation of quantum mechanics was developed by Paul Dirac, David Hilbert, John von Neumann, and Hermann Weyl.

2

, 1x t dx

*

1 2 1 2 1 2, , ,x t x t x x dx dx

[ the necessary condition: |Ψ(x, t)|2 →0 as x → ±∞ ]

Basic postulates of Schrődinger’s Wave Mechanics -II

5. The Principle of Superposition: If Ψ1(x, t) is a state with an observable value L1 and Ψ2(x, t) is another state of the same system with the observable value L2, then any linear combination,

Ψ(x, t) = c1Ψ1(x, t) + c2Ψ2(x, t),

is a state for which the same measurement results in either L1 or L2 with the respective probabilities |c1|2 and |c2|2. Ψ(x, t) is a solution of a linear equation.

6. Any physical observable is associated with a self-adjoint linear operator. The operators must yield real eigenvalues. Operators must be Hermitian.

7. The Correspondence Principle (N. Bohr, 1920): quantum mechanical description of systems must reproduce predictions of classical mechanics in the limit of large quantum numbers or, equivalently, when ħ → 0. For large systems, the quantum mechanical description must closely approximate the classical description

1 1 1ˆ ,F x K x x x dx

Time-dependent Schrӧdinger Equation in 1D case. Separation of variables.

8. For any conservative system, its wave function is a solution of Schrӧdinger’s Equation:

Introducing the operators: (linear momentum) and , Schrӧdinger’s Equation can be presented as

2 2

2

2i V x

t m x

p i x x x

ˆi Ht

with

2ˆˆ2

pH V x

m

The Hamiltonian is the operator of total energy

Time-dependent Schrődinger equation

Time-dependent Schrődinger equation

Separation of variables: ,x t x t

ˆi x t t H xt

Left-multiplying each side by gives:

1

x t

1 ˆit H x

t t x

Function of t Function of x

Stationary Schrӧdinger Equation

/iEt

it E

t t

i t E tt

t Ae

1 ˆ

ˆ

ˆ

*

*

H x Ex

H x E x

x H x dx E x x dx

If * 1x x dx

then

ˆ* x H x dx E

E is the average value of the Hamiltonian, H

Stationary Schrӧdinger Equation

/, iEtx t Ae x

Example: Simple Harmonic Motion (the harmonic oscillator problem) 2 2

2

2

2 22

2

1ˆ2 2

ˆ

1

2 2

dH kx

m dx

H E

dkx E

m dx

2

12

/

, n

m x

n n

kE n

m

mx Ae H x

A Brief Review

• Time-dependent Schrődinger Equation:

• Meaning of operators in quantum theory:

• Meaning of the wave function:

• If the Hamiltonian, Ĥ, is not a function of time, energy conserves, and the time-dependent Schrődinger Equation is separable:

ˆi Ht

2ˆˆ2

pH V x

m

is the operator of total energy, the Hamiltonian

ˆd

p idx

Operator of linear momentum of the moving particle

* ˆ, ,A t x t A x t dx

is the (statistical ) average of the respective observable value

2

,x t dx probability of finding the particle in the interval (x, x + dx) at time t

/ ˆ, , where iEtx t Ae x H x E x

An Important Example: Simple Harmonic Motion (the problem of a harmonic oscillator)

Solving the stationary Schrődinger equation,

with

Then the Schrődinger equation becomes

At , we have: So with Therefore, substitute

into . This gives:

with

Its solution is

are Hermite polynomials. Thus

2ˆˆ

2

pH V x

m

22 2

2

2 22

2

,

1ˆ2 2

d dp i p

dx dx

dH kx

m dx

2 22

2

1

2 2

dkx E

m dx

x 2 2

2

2

10

2 2

dkx

m dx

2 /2m xx e

H x E x

2 /2m xx e x

2 22

2

1

2 2

dkx E

m dx

k

m

2

2 20

m mE m

const as x x

12

, , 0,1,2,3,...n n nx AH x E n n

nH x 2 /2m x

n nx Ae H x

Harmonic Oscillators in a Solid Matter

Boltzmann’s probability of finding an oscillating atom in nth excited state is proportional to

1 12 2

//nn kT nE kT

e e e

Average energy is

00

0 0

with 1

nn

n n

EE

nnn

E E

n n

eE e

Ek

eT

e

As , we have: 12nE n /2 /2

0 0 0

/2 /2 /2

/1 1 1

nn

E n

n n n

kT

kT

e e e e e

e e e

e e e

Plank’s formula for black-body radiation:

2

5 /

2

1hc kT

c hI

e

2

c

2

h

Hilbert Space as a Tool Solving the stationary Schrődinger equation,

We come to a complete set of eigenfunctions, , that can be used as a basis set for any other function:

Plugging it into the original equation, it can be converted into the algebraic eigenvalue problem

Here Ĥ is a matrix with matrix elements

For a solid state, its oscillator states are:

H x E x

n x

n n

n

x xc

H Ec c

ˆ*mnH x H x dx

1 1 2 2 2 3 4 4 ...n n n nx x x x