Operators and Operator in Quantum

Alexander Dzyubenko

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

Supported in part by NSF Department of Mathematics, CSUB September 22, 2004

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v Quantum Harmonic Oscillator

Ø Schrödinger Equation (SE)

Ladder Operators

Ø Coherent and Squeezed Oscillator States

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Total Energy = Kinetic Energy + Potential Energy

p2 mw 2 x2 H = K +V (x) = + 2m 2 Hamiltonian Function

k

Frequency m k 2p w = = m T Total Energy: E = mw2 A2 Period Amplitude

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Total Energy = Kinetic Energy + Potential Energy

pˆ 2 mw2 xˆ2 Hˆ = Kˆ +Vˆ = + 2m 2

Probability Hamiltonian Operator: Acts on Wave Functions Y = Y(x,t) = x Y Density: 2 ¶ | Y(x,t) | Momentum Operator pˆ = -ih ¶x Coordinate Operator xˆ = x

Time-Dependent SchroedingerEquation: ¶Y(x,t) ih = HˆY(x,t) ¶t

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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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Time-Dependent SchroedingerEquation (SE): ¶Y(x,t) ih = HˆY(x,t) = EY(x,t) ¶t

E Energy Y(x,t) = j(x)exp(-i t) := j(x)exp(-iwt) EigenvalueE h

Time-Independent SE: Hˆj(x) = Ej(x)

h2 d 2j mw2 x2 For a Harmonic Oscillator: - + j = Ej 2m dx2 2

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d 2j 2m æ mw2 x2 ö + ç E - ÷j = 0 - ¥ < x < +¥ 2 2 ç ÷ dx h è 2 ø

• A well-known equation in mathematics

• Solutions were explored by Charles Hermite(1822-1901)

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Physically reasonable wave functions must drop to zero for | x |® ¥

Symmetry of the potential => even and odd solutions Potential Barrier

Discrete Energy

EigenvaluesEn QM: Eigenfunctions: are only allowed Penetration QM Wavefunctions “Below the Barrier”

PDF created with pdfFactory Pro trial version www.pdffactory.com (Quantum) Operator Approach: Non-Commutativity, Ladder Operators, …

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¶ Momentum Operator pˆ = -ih ¶x Coordinate Operator xˆ = x

Commutator: [ pˆ, xˆ] = pˆxˆ - xˆpˆ = -ih

d df [ pˆ, xˆ] f = -ih ( fx) + ih x = -ihf " f (x) dx dx

Non-Commutativity=> Heisenberg Uncertainty Principle

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HermitianConjugates:

* y Aˆ j = ( j Aˆ + y ) or, incoordinate representation, * ò dxy (x)* Aˆ(x)j(x) = (ò dxj(x)* Aˆ + (x)y (x))

HermitianOperators: Unitary Operators: ˆ ˆ + A = A Uˆ + = Uˆ -1

Real Eigenvalues Describe Time Evolution in QM Complete Set of Eigenstates Perform Operator Transformations Physical Observables in QM are Aˆ ® Bˆ = UˆAˆUˆ + described by HermitianOperators

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Non-Commutativity: [ pˆ, xˆ] = pˆxˆ - xˆpˆ = -ih

Heisenberg Uncertainty Principle: Dp × Dx ³ h / 2

2 DA = Aˆ 2 - Aˆ Aˆ = j Aˆ j = ò dxj(x)* Aˆ(x)j(x) Uncertainty= Mean Square Deviation Expectation Value of A = Sqrt[Variance]

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

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mw Xˆ = xˆ = X h Dimensionless [Pˆ, Xˆ ] = -i Operators pˆ d Pˆ = = -i mwh dX

Ladder Operators Non-commutative 1 1 æ d ö a = (Xˆ + iPˆ )= ç X + ÷ : 2 2 è dX ø +

+ 1 1 æ d ö [a,a ] = 1 a = (Xˆ - iPˆ)= ç X - ÷ 2 2 è dX ø

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pˆ 2 mw2 xˆ 2 Hˆ = + = hw(a+a + 1 ) 2m 2 2

algebra: [a,a + ] = 1

Lowering Raising

a 0 = 0 Coordinate Representation: X 0 = exp(- 1 X 2 ) Vacuum State 2 1 æ d ö a = ç X + ÷ Gaussian 2 è dX ø

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1 n n = (a+ ) 0 n! Excited States n = 1, 2, 3, …

+ algebra Raising a n = n +1 n +1 => [a,a+ ] = 1 Lowering a n = n n -1

+ Particle Number Operator a a n = n n

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Probability Distributions

+ ˆ 1 1 + n H = hw(a a + 2 ) n = (a ) 0 n! Hamiltonian (Energy Operator) n = 0, 1, 2, …

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

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

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Displacement Operator Dˆ (a) = exp(aa + -a *a)

Complex number Unitary

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

Minimum Behave Uncertainty Dp × Dx = h / 2 “most classically” States

¥ n 1 2 a a = exp(- 2 |a | )å n n=0 n!

PDF created with pdfFactory Pro trial version www.pdffactory.com Disentangling the Operators

ˆ + * Displacement D(a) = exp(aa -a a) = Operator 1 2 + * = exp(- 2 |a | )exp(aa )exp(-a a)

Baker-Campbell- Algebra + Hausdorfformula Used: [a,a ] = 1

Coherent States a = Dˆ (a) 0 Superposition of an infinite # of oscillator ¥ a n states |n> with proper 1 2 amplitudes and phases a = exp(- 2 |a | )å n n=0 n!

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Dˆ + (a)a+ Dˆ (a) = a+ + a Dˆ (a) = exp(aa+ -a *a) Dˆ + (a)aDˆ (a) = a + a *

Generates an d Rea d Ima Overcomplete Set a a = 1ˆ of Coherent States ò p

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

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2 ˆ 1 + 1 * 2 Unitary S(b ) = exp(2 ba - 2 b a )

Mixes Creation and Annihilation Operators: Sˆ + (b )a +Sˆ(b ) = cosh(b )a + - sinh(b )a Sˆ + (b )a Sˆ(b ) = cosh(b )a - sinh(b )a+

Generates Squeezed States: b = Sˆ(b ) 0

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

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Bose-Einstein Statistics: + + + Commutation Relation [a,a ] = aa - a a = 1

Bosons: photons, phonons, …

Squeezed States: Theory of Superfluidity

Fermi-DiracStatistics: + + + AntiCommutation {a,a } = aa + a a = 1 Relation Fermions: electrons, protons, …

Squeezed States: Theory of Superconductivity

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It works nicely and not just for tiny particles

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

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

No deviations from quantum mechanics are known

A HUGE problem: to describe gravitation in the framework of quantum mechanics “Theory of everything” will be a relativistic quantum-mechanical theory

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