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Operator Algebras in Quantum Mechanics

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Operator Algebras in Quantum Mechanics Operators and Operator Algebras in Quantum Mechanics Alexander Dzyubenko Department of Physics, California State University at Bakersfield Department of Physics, University at Buffalo, SUNY Supported in part by NSF Department of Mathematics, CSUB September 22, 2004 PDF created with pdfFactory Pro trial version www.pdffactory.com Outline v Quantum Harmonic Oscillator Ø Schrödinger Equation (SE) Ladder Operators Ø Coherent and Squeezed Oscillator States PDF created with pdfFactory Pro trial version www.pdffactory.com Classical Harmonic Oscillator 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 PDF created with pdfFactory Pro trial version www.pdffactory.com Quantum Harmonic Oscillator 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 PDF created with pdfFactory Pro trial version www.pdffactory.com 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 PDF created with pdfFactory Pro trial version www.pdffactory.com SE as an EigenvalueEquation 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 PDF created with pdfFactory Pro trial version www.pdffactory.com SE for Quantum Harmonic Oscillator 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) PDF created with pdfFactory Pro trial version www.pdffactory.com Requirements for a Solution 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, … PDF created with pdfFactory Pro trial version www.pdffactory.com Non-Commutative Algebras ¶ 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 PDF created with pdfFactory Pro trial version www.pdffactory.com Hermitianand Unitary Operators 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 PDF created with pdfFactory Pro trial version www.pdffactory.com Heisenberg Uncertainty Principle 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 PDF created with pdfFactory Pro trial version www.pdffactory.com Non-Commutative Ladder Operators 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 + ÷ algebra: 2 2 è dX ø + + 1 1 æ d ö [a,a ] = 1 a = (Xˆ - iPˆ)= ç X - ÷ 2 2 è dX ø PDF created with pdfFactory Pro trial version www.pdffactory.com Hamiltonian and Ladder Operators 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 PDF created with pdfFactory Pro trial version www.pdffactory.com (Bose) Ladder Operators 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 PDF created with pdfFactory Pro trial version www.pdffactory.com Hamiltonian, Ladder Operators, Eigenstates Equidistant Energy Levels 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 PDF created with pdfFactory Pro trial version www.pdffactory.com Oscillator Eigenstates: ProbabiltyDistribution http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/ PDF created with pdfFactory Pro trial version www.pdffactory.com Oscillator Coherent States 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! PDF created with pdfFactory Pro trial version www.pdffactory.com Displacement Operator 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 PDF created with pdfFactory Pro trial version www.pdffactory.com Squeezing Operator 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 PDF created with pdfFactory Pro trial version www.pdffactory.com Bose and Fermi Statisics 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 PDF created with pdfFactory Pro trial version www.pdffactory.com How successful is quantum mechanics? 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 PDF created with pdfFactory Pro trial version www.pdffactory.com.
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