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 motion in order to determine the stationary states of the quantum harmonic oscillator which has a quantum Hamiltonian of the form: $$ \mathbf{H} = (\frac {1}{2m})\cdot P^2 + \frac{k}{2}\cdot X^2~, $$ where $$X$$ and $$P$$ are, respectively, the coordinate and conjugate momentum operators. $$X$$ and $$P$$ satisfy the Heisenberg commutation/'uncertainty' relations $$[X,P] = i\hbar I~,$$ where the 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: $$ \mathbf{x} = (\frac{X}{\alpha})~, ~ \mathbf{p}= (\frac{\alpha P}{\hbar}) ~=and= ~ \alpha = \sqrt {\frac{\hbar}{mk}}~. $$ With these new dimensionless operators, $$\mathbf{x}$$ and $$\mathbf{p}$$, the quantum Hamiltonian takes the form: $$ \mathbf{H}= (\frac{\hbar \omega}{2})\cdot (\mathbf{p}^2 + \mathbf{x}^2)~, $$ which in units of $$\hbar \cdot \omega$$ is simply: $$ \mathbf{H}' = (\frac {1}{2})\cdot (\mathbf{p}^2 + \mathbf{x}^2)~. $$ The commutator of $$\mathbf{x}$$ with its conjugate operator $$\mathbf{p}$$ is simply $$[\mathbf{x}, \mathbf{p}] = i$$~.\\

Next one defines the superoperators S_{Hx} = [H, x] = -i \cdot p$$, and $$S_{Hp} = [H, p] = i \cdot \mathbf{x}$$ that will lead to new operators that act as generators of a Lie Algebra for this quantum harmonic oscillator. The eigenvectors Z of these superoperators are obtained by solving the equation $$S_H \cdot Z = \zeta Z$$, where $$\zeta$$ are the eigenvalues, and $$Z can be written as $$(c_1 \cdot x + c_2 \cdot p)$$~. 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: $$ a^\dagger = c_1* (x-ip)~, =and= ~ a = c_1 (x+ip)~, $$ respectively for $$\zeta = \pm 1$$~. For $$c_1 =\surd {2}$$ one obtains normalized operators $$H, a$$ and $$a \dagger$$ that generate a $$4$$--dimensional Lie algebra with commutators: $$ [H,a] = -a~,~[H, a^\dagger]= a^\dagger~, ~ =and= ~ [a, a^\dagger]= I ~. $$ The term $$\mathbf{a}$$ is called the annihilation operator and the term $$a\dagger$$ is called the creation operator. This Lie algebra is solvable and generates after repeated application of $$a\dagger$$ all the eigenvectors of the quantum harmonic oscillator: $$ \Phi_n = (\frac{(a\dagger)^n}{\surd(n!)})\cdot \Phi_0 ~. $$ The corresponding, possible eigenvalues for the energy, derived then as solutions of the Schr\"odinger equations for the quantum harmonic oscillator are: $$ E_n = \hbar \cdot \omega (n+ \frac{1}{2}) ~, ~=where= ~ n = 0,1, \ldots, N~. $$ The position and momentum eigenvector coordinates can be then also computed by iteration from (finite) matrix representations of the (finite)  Lie algebra, using, for example, a simple 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 Hermite polynomials in terms of the coordinates. In the coordinate representation the quantum Hamiltonian and bosonic  operators have, respectively, the simple expressions:

$$

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})~.

$$ The ground state eigenfunction normalized to unity is obtained from solving the simple first-order differential equation $$a\Phi_0 (x) = 0$$ and which leads to the expression: $$ \Phi_0 (x)= (\pi^{-\frac{1}{4}})\cdot \exp(-\frac{x^2}{2})~. $$ By repeated application of the creation operator written as $$ a\dagger = (-\frac{1}{\surd 2})\cdot (\exp(\frac{x^2}{2}))\cdot(\frac{d}{dx^2})\cdot \exp(-\frac{x^2}{2}) ~, $$ one obtains the $$n$$-th level eigenfunction: $$ \Phi_n(x) = (\frac{1}{(\surd\pi) 2^n n!)})\cdot (\mathbf{He}_n (x))~, $$ where $$\mathbf{He}_n(x)$$ is the Hermite polynomial of order $$n$$~. With the special generating function of the Hermite polynomials $$ 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 \Phi_n(x) ~. $$ Such applications of the Lie algebra, and the related algebra of the bosonic operators as defined above are quite numerous in theoretical physics, and especially for various quantum field carriers in QFT that are all bosons. (Please note also the additional examples of special `Lie' superalgebras for gravitational and other fields, related to hypothetical particles such as gravitons and Goldstone quanta that are all bosons of different spin values and `Penrose homogeneity' ).\\

In the interesting case of a two-mode bosonic quantum system formed by the 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 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 computations of vibrational (IR, Raman, etc.) spectra of two--mode, diatomic 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).