PlanetPhysics/Quantum Harmonic Oscillator and Lie Algebra

Lie Algebra of a Quantum Harmonic Oscillator

One wishes to solve the time-independent Schr\"odinger equation of [[../CosmologicalConstant/|motion]] in order to determine the stationary states of the quantum harmonic oscillator which has a quantum [[../Hamiltonian2/|Hamiltonian]] of the form:

$𝐇=(\frac{1}{2m})\cdot P^2+\frac{k}{2}\cdot X^2,$

where $X$ and $P$ are, respectively, the coordinate and conjugate [[../Momentum/|momentum]] [[../QuantumOperatorAlgebra4/|operators]]. $X$ and $P$ satisfy the Heisenberg commutation/'uncertainty' [[../Bijective/|relations]] $[X,P]=i\hslash I,$ where the [[../QuantumParticle/|identity operator]] $I$ is employed to simplify notation. A simpler, equivalent form of the above Hamiltonian is obtained by defining physically dimensionless coordinate and momentum:

$𝐱=(\frac{X}{\alpha }),𝐩=(\frac{\alpha P}{\hslash })=and=\alpha =\sqrt{\frac{\hslash }{mk}}.$

With these new dimensionless operators, $𝐱$ and $𝐩$, the quantum Hamiltonian takes the form:

$𝐇=(\frac{\hslash \omega }{2})\cdot ({𝐩}^2+{𝐱}^2),$

which in units of $\hslash \cdot \omega$ is simply:

${𝐇}^{\prime }=(\frac{1}{2})\cdot ({𝐩}^2+{𝐱}^2).$

The [[../Commutator/|commutator]] of $𝐱$ with its conjugate [[../QuantumSpinNetworkFunctor2/|operator]] $𝐩$ is simply $[𝐱,𝐩]=i$~.\\

Next one defines the superoperators </math>S_{Hx} = [H, x] = -i \cdot p$,and$S_{Hp} = [H, p] = i \cdot \mathbf{x}$thatwillleadtonewoperatorsthatactas[[../Generator/|generators]]ofa[[../TopologicalOrder2/|LieAlgebra]]forthisquantumharmonicoscillator.Theeigenvectors<math>Z$ of these superoperators are obtained by solving the equation </math>S_H \cdot Z = \zeta Z$,where$\zeta$aretheeigenvalues,and$Z$canbewrittenas<math>(c_1\cdot x+c_2\cdot p)$~. The solutions are

$\zeta =\pm 1,=and=c_2=\mp i\cdot c_1.$

Therefore, the two eigenvectors of $S_H$ can be written as:

$a^{†}=c_1*(x-ip),=and=a=c_1(x+ip),$

respectively for $\zeta =\pm 1$~. For $c_1=\surd 2$ one obtains normalized operators $H,a$ and $a†$ that generate a $4$--dimensional Lie algebra with commutators:

$[H,a]=-a,[H,a^{†}]=a^{†},=and=[a,a^{†}]=I.$

The term $𝐚$ is called the annihilation operator and the term $a†$ is called the creation operator. This Lie algebra is solvable and generates after repeated application of $a†$ all the eigenvectors of the quantum harmonic oscillator:

${\mathrm{\Phi }}_n=(\frac{(a†{)}^n}{\surd (n!)})\cdot {\mathrm{\Phi }}_0.$

The corresponding, possible eigenvalues for the [[../CosmologicalConstant/|energy]], derived then as solutions of the Schr\"odinger equations for the quantum harmonic oscillator are:

$E_n=\hslash \cdot \omega (n+\frac{1}{2}),=where=n=0,1,\dots ,N.$

The [[../Position/|position]] and momentum eigenvector coordinates can be then also computed by iteration from (finite) [[../Matrix/|matrix]] [[../CategoricalGroupRepresentation/|representations]] of the (finite) Lie algebra, using, for example, a simple [[../SupercomputerArchitercture/|computer]] programme to calculate linear expressions of the annihilation and creation operators. For example, one can show analytically that:

$[a,x^k]=(\frac{k}{\surd 2})\cdot (x_{k-1}).$

One can also show by introducing a coordinate representation that the eigenvectors of the harmonic oscillator can be expressed as [[../HermitePolynomials/|Hermite polynomials]] in terms of the coordinates. In the coordinate representation the quantum Hamiltonian and bosonic operators have, respectively, the simple expressions:

Failed to parse (syntax error): {\displaystyle H &= (\frac{1}{2})\cdot[-\frac{d^2}{dx^2}) + (x^2)]~, \\ a &= (\frac{1}{\surd 2})\cdot (x + \frac{d}{dx})~, \\ a\dagger &= (\frac{1}{\surd 2})\cdot (x - \frac{d}{dx})~. }

The ground state eigenfunction normalized to unity is obtained from solving the simple first-order [[../DifferentialEquations/|differential equation]] $a{\mathrm{\Phi }}_0(x)=0$ and which leads to the expression:

${\mathrm{\Phi }}_0(x)=({\pi }^{-\frac{1}{4}})\cdot \exp (-\frac{x^2}{2}).$

By repeated application of the creation operator written as

$a†=(-\frac{1}{\surd 2})\cdot (\exp (\frac{x^2}{2}))\cdot (\frac{d}{d{x}^2})\cdot \exp (-\frac{x^2}{2}),$

one obtains the $n$-th level eigenfunction:

${\mathrm{\Phi }}_n(x)=(\frac{1}{(\surd \pi )2^{n}n!)})\cdot ({𝐇𝐞}_n(x)),$

where ${𝐇𝐞}_n(x)$ is the Hermite polynomial of order $n$~. With the special generating [[../Bijective/|function]] of the Hermite polynomials

$F(t,x)=({\pi }^{-\frac{1}{4}})\cdot (\exp ((-\frac{x^2}{2})+tx-(\frac{t^2}{4})),$

one obtains explicit analytical relations between the eigenfunctions of the quantum harmonic oscillator and the above special generating function:

$F(t,x)=\sum _{n=0}(\frac{t^n}{\surd (2^n\cdot n!)})\cdot {\mathrm{\Phi }}_n(x).$

Such applications of the [[../LieAlgebra/|Lie algebra]], and the related algebra of the bosonic operators as defined above are quite numerous in [[../PhysicalMathematics2/|theoretical physics]], and especially for various [[../CosmologicalConstant2/|quantum field]] carriers in [[../HotFusion/|QFT]] that are all [[../AntiCommutationRelations/|bosons]]. (Please note also the additional examples of special `[[../AntiCommutationRelations/|Lie' superalgebras]] for gravitational and other [[../CosmologicalConstant2/|fields]], related to hypothetical particles such as [[../BoseEinsteinStatistics/|gravitons]] and Goldstone quanta that are all bosons of different [[../QuarkAntiquarkPair/|spin]] values and `Penrose homogeneity' ).\\

In the interesting case of a two-mode bosonic quantum [[../SimilarityAndAnalogousSystemsDynamicAdjointnessAndTopologicalEquivalence/|system]] formed by the [[../Tensor/|tensor]] (direct) product of one-mode bosonic states: $\mid m,n>:=\mid m>\otimes \mid n>$, one can generate a $3$--dimensional Lie algebra in terms of Casimir operators. Finite -- dimensional [[../BilinearMap/|Lie algebras]] are far more tractable, or easier to compute, than those with an infinite basis set. For example, such a Lie algebra as the $3$--dimensional one considered above for the two-mode, bosonic states is quite useful for numerical [[../LQG2/|computations]] of vibrational ([[../SpectralImaging/|IR]], [[../FluorescenceCrossCorrelationSpectroscopy/|Raman]], etc.) spectra of two--mode, diatomic [[../Molecule/|molecules]], as well as the computation of scattering states. Other perturbative calculations for more complex quantum systems, as well as calculations of exact solutions by means of Lie algebras have also been developed (see for example Fernandez and Castro,1996).