Operators and Operator Algebras in Quantum Mechanics - CSU Bakersfield

Operators and Operator Algebras in Quantum Mechanics

Alexander Dzyubenko

Department of Physics, California State University at BakersfieldDepartment of Physics, University at Buffalo, SUNY

Department of Mathematics, CSUB September 22, 2004

Supported in part by NSF

PDF created with pdfFactory Pro trial version www.pdffactory.com

http://www.pdffactory.com

Outline

v Quantum Harmonic Oscillator

Ladder Operators

Ø Schrödinger Equation (SE)

Ø Coherent and Squeezed Oscillator States



Hamiltonian Function

Classical Harmonic Oscillator

22)(

222 xmm

pxVKH ω+=+=

Total Energy = Kinetic Energy + Potential Energy

22 AmE ω=Total Energy:Amplitude

Frequency

Period

Tmk π

ω2

==

mk



Hamiltonian Operator: Acts on Wave Functions

Quantum Harmonic Oscillator

2ˆ

2ˆˆˆˆ

222 xmm

pVKH ω+=+=

Total Energy = Kinetic Energy + Potential Energy

xip

∂∂

−= hˆMomentum Operator

xx =ˆCoordinate Operator

Ψ=Ψ=Ψ xtx ),(

Time-Dependent Schroedinger Equation:

),(ˆ),( txHt

txi Ψ=∂

Ψ∂h

ProbabilityDensity:

2|),(| txΨ



Erwin Schrödinger Once at the end of a colloquium

I heard Debye saying something like: “Schrödinger, you are not working right now on very important problems… why don’t you tell us some time about that thesis of de Broglie’s…

In one of the next colloquia, Schrödinger gave a beautifully clear account of how de Broglie associated a wave with a particle, and how he could obtain the quantization rules…

When he had finished, Debye casually remarked that he thought this way of talking was rather childish… To deal properly with waves, one had to have a wave equation.

Felix Bloch

Address to the American Physical Society, 1976



Energy Eigenvalue E)exp()(:)exp()(),( tixtEixtx ωϕϕ −=−=Ψ

h

Time-Dependent Schroedinger Equation (SE):

),(),(ˆ),( txEtxHt

txi Ψ=Ψ=∂

Ψ∂h

SE as an Eigenvalue Equation

Time-Independent SE:

)()(ˆ xExH ϕϕ =

For a Harmonic Oscillator: ϕϕωϕ Exm

dxd

m=+−

22

22

2

22h



• A well-known equation in mathematics

• Solutions were explored byCharles Hermite (1822-1901)

02

2 22

22

2

=

−+ ϕ

ωϕ xmEmdxd

h

SE for Quantum Harmonic Oscillator

+∞<<∞− x



Physically reasonable wave functions must drop to zero for ∞→|| x

Symmetry of the potential => even and odd solutions

Requirements for a Solution

Potential Barrier

Eigenfunctions:QM Wavefunctions

QM: Penetration “Below the

Barrier”

Discrete Energy Eigenvalues En are only allowed



(Quantum) Operator Approach:Non-Commutativity, Ladder Operators, …



Non-Commutative Algebras

xip

∂∂

−= hˆMomentum Operator

xx =ˆCoordinate Operator

)()(]ˆ,ˆ[ xffixdxdfifx

dxdifxp ∀−=+−= hhh

hipxxpxp −=−= ˆˆˆˆ]ˆ,ˆ[Commutator:

Non-Commutativity => Heisenberg Uncertainty Principle



( )

( )***

*

)()(ˆ)()()(ˆ)(

,,

ˆˆ

xxAxdxxxAxdx

tionrepresentacoordinateinorAA

ψϕϕψ

ψϕϕψ

+

+

∫∫ =

=

Hermitian Conjugates:

+= AA ˆˆHermitian Operators:

1ˆˆ −+ = UUUnitary Operators:

Real EigenvaluesComplete Set of Eigenstates

Physical Observables in QM are described by Hermitian Operators

Describe Time Evolution in QMPerform Operator Transformations

+=→ UAUBA ˆˆˆˆˆ

Hermitian and Unitary Operators



Heisenberg Uncertainty Principle

hipxxpxp −=−= ˆˆˆˆ]ˆ,ˆ[Non-Commutativity:

22 ˆˆ AAA −=∆

Uncertainty= Mean Square Deviation= Sqrt[Variance]

Heisenberg Uncertainty Principle:

Physical Observable A => Hermitian (self-adjoint) operator A

2/h≥∆⋅∆ xp

)()(ˆ)(ˆˆ * xxAxdxAA ϕϕϕϕ ∫==

Expectation Value of A



Non-Commutative Ladder Operators

Dimensionless Operators

Ladder Operators

dXdi

mpP −==

hωˆˆ

XxmX == ˆˆh

ω

( )

+=+=

dXdXPiXa

21ˆˆ

21

iXP −=]ˆ,ˆ[

( )

−=−=+

dXdXPiXa

21ˆˆ

21 1],[ =+aa

Non-commutative algebra:



Hamiltonian and Ladder Operators

Lowering

1],[ =+aa

( )21

222

2ˆ

2ˆˆ +=+= +aaxmm

pH ωω

h

algebra:

Raising

00 =a

Vacuum State( )2

21exp0 XX −=

+=

dXdXa

21

Coordinate Representation:

Gaussian



(Bose) Ladder Operators

( ) 0!

1 nan

n +=

Excited States n = 1, 2, 3, …

algebra

1],[ =+aa

11 ++=+ nnnaRaising

Lowering 1−= nnna

Particle Number Operator nnnaa =+

=>



( )21ˆ += +aaH ωh ( ) 0

!1 nan

n +=

Hamiltonian, Ladder Operators, Eigenstates

n = 0, 1, 2, …Hamiltonian

(Energy Operator)

Equidistant Energy Levels

http://vega.fjfi.cvut.cz/docs/pvok/aplety/kv_osc/index.html

Probability Distributions



Oscillator Eigenstates: Probabilty Distribution

http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/



Oscillator Coherent States

( )aaD *exp)(ˆ ααα −= +

Complex number

Displacement Operator

0)(ˆ αα D=Coherent States Schrödinger 1927 (in a different form)

( ) nnn

n

∑∞

=

−=0

221

!||exp α

αα

Minimum Uncertainty States

2/h=∆⋅∆ xp Behave “most classically”

Unitary



Disentangling the Operators

( )( ) ( ) ( )aa

aaD*2

21

*

expexp||expexp)(ˆ

ααα

ααα

−−=

=−=+

+Displacement Operator

0)(ˆ αα D=Coherent States

( ) nnn

n

∑∞

=

−=0

221

!||exp α

αα

1],[ =+aaAlgebraUsed:

Superposition of an infinite # of oscillator states |n> with proper amplitudes and phases

Baker-Campbell-Hausdorf formula



Displacement Operator

ααα += +++ aDaD )(ˆ)(ˆ

*)(ˆ)(ˆ ααα +=+ aDaD( )aaD *exp)(ˆ ααα −= +

Generates an Overcomplete Set of Coherent States

1ImRe=∫ αα

παα dd

Very useful in Quantum Optics and Field Theory 0)(ˆ αα D=



Squeezing Operator

( )2*212

21exp)(ˆ aaS βββ −= +

Mixes Creation and Annihilation Operators:

aaSaS )sinh()cosh()(ˆ)(ˆ ββββ −= +++

++ −= aaSaS )sinh()cosh()(ˆ)(ˆ ββββ

Minimum Uncertainty States + One coordinate uncertainty E.g., ∆x can be made arbitrarily small (squeezed) at the expense of ∆p. Reduction of Quantum Noise

Generates Squeezed States: 0)(ˆ ββ S=

Unitary



Bose and Fermi Statisics

1],[ =−= +++ aaaaaaBose-Einstein Statistics: Commutation Relation

Bosons: photons, phonons, …

1},{ =+= +++ aaaaaaFermi-Dirac Statistics: AntiCommutationRelation

Fermions: electrons, protons, …

Squeezed States: Theory of Superfluidity

Squeezed States: Theory of Superconductivity



How successful is quantum mechanics?

It works nicely and not just for tiny particles

No deviations from quantum mechanics are known

“Theory of everything” will be a relativistic quantum-mechanical theory

QM is crucial to explain how the Universe formed QM “explains” the stability of matter

A HUGE problem: to describe gravitation in the framework of quantum mechanics

QM explains Macroscopic Quantum Phenomena: Superconductivity, Superfluidity, Josephson Effect, and the Quantum Hall Effect



Documents

Operators and Operator Algebras in Quantum Mechanics - CSU Bakersfield