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Operators and Operator Algebras in Quantum Mechanics - CSU Bakersfield
Transcript of 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
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Outline
v Quantum Harmonic Oscillator
Ladder Operators
Ø Schrödinger Equation (SE)
Ø Coherent and Squeezed Oscillator States
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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
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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
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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
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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
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• 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
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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
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(Quantum) Operator Approach:Non-Commutativity, Ladder Operators, …
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Non-Commutative Algebras
xip
∂∂
−= hˆMomentum Operator
xx =ˆCoordinate Operator
)()(]ˆ,ˆ[ xffixdxdfifx
dxdifxp ∀−=+−= hhh
hipxxpxp −=−= ˆˆˆˆ]ˆ,ˆ[Commutator:
Non-Commutativity => Heisenberg Uncertainty Principle
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( )
( )***
*
)()(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
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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
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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:
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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
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(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 =+
=>
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( )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
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Oscillator Eigenstates: Probabilty Distribution
http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/
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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
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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
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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=
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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
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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
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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
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