Quantum harmonic oscillator

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The quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary potential can be approximated as a harmonic potential at the vicinity of a stable equilibrium point, it is one of the most important model systems in quantum mechanics. Furthermore, it is one of the few quantum-mechanical systems for which an exact, analytical solution is known.^[1] ^[2] ^[3]

1 One-dimensional harmonic oscillator
2 N-dimensional harmonic oscillator
- 2.1 Example: 3D isotropic harmonic oscillator
3 Coupled harmonic oscillators
4 See also
5 References
6 External links

One-dimensional harmonic oscillator

Hamiltonian and energy eigenstates

The Hamiltonian of the particle is:

$\hat H = \frac{{\hat p}^2}{2m} %2B \frac{1}{2} m \omega^2 {\hat x}^2 \, ,$

where $\hat x = x$ is the position operator, and $\hat p$ is the momentum operator, given by

$\hat p = - i \hbar {\partial \over \partial x} \, .$

The first term in the Hamiltonian represents the kinetic energy of the particle, and the second term represents the potential energy in which it resides. In order to find the energy levels and the corresponding energy eigenstates, we must solve the time-independent Schrödinger equation,

$\hat H \left| \psi \right\rangle = E \left| \psi \right\rangle \, .$

We can solve the differential equation in the coordinate basis, using a spectral method. It turns out that there is a family of solutions. In the position basis they are

$\psi_n(x) = \sqrt{\frac{1}{2^n\,n!}} \cdot \left(\frac{m\omega}{\pi \hbar}\right)^{1/4} \cdot e^{ - \frac{m\omega x^2}{2 \hbar}} \cdot H_n\left(\sqrt{\frac{m\omega}{\hbar}} x \right), \qquad n = 0,1,2,\ldots.$

The functions H_n are the Hermite polynomials:

$H_n(x)=(-1)^n e^{x^2}\frac{d^n}{dx^n}\left(e^{-x^2}\right)$

The corresponding energy levels are

$E_n = \hbar \omega \left(n %2B {1\over 2}\right)$ .

This energy spectrum is noteworthy for three reasons. First, the energies are quantized, meaning that only discrete energy values (half-integer multiples of $\hbar\omega$ ) are possible; this is a general feature of quantum-mechanical systems when a particle is confined. Second, these discrete energy levels are equally spaced, unlike in the Bohr model of the atom, or the particle in a box. Third, the lowest achievable energy (the energy of the $n = 0$ state, called the ground state) is not equal to the minimum of the potential well, but $\hbar\omega/2$ above it; this is called zero-point energy. Because of the zero-point energy, the position and momentum of the oscillator in the ground state are not fixed (as they would be in a classical oscillator), but have a small range of variance, in accordance with the Heisenberg uncertainty principle. The zero-point energy also has important implications in quantum field theory and quantum gravity.

Note that the ground state probability density is concentrated at the origin. This means the particle spends most of its time at the bottom of the potential well, as we would expect for a state with little energy. As the energy increases, the probability density becomes concentrated at the classical "turning points", where the state's energy coincides with the potential energy. This is consistent with the classical harmonic oscillator, in which the particle spends most of its time (and is therefore most likely to be found) at the turning points, where it is the slowest. The correspondence principle is thus satisfied.

Ladder operator method

The spectral method solution, though straightforward, is rather tedious. The "ladder operator" method, due to Paul Dirac, allows us to extract the energy eigenvalues without directly solving the differential equation. Furthermore, it is readily generalizable to more complicated problems, notably in quantum field theory. Following this approach, we define the operators a and its adjoint a^†

$\begin{align} a &=\sqrt{m\omega \over 2\hbar} \left(x %2B {i \over m \omega} p \right) \\ a^{\dagger} &=\sqrt{m \omega \over 2\hbar} \left( x - {i \over m \omega} p \right) \end{align}$

The operator a is not Hermitian since it and its adjoint a^† are not equal.

We can also define a number operator N which has the following property:

$N = a^{\dagger}a$

$N\left| n \right\rangle =n\left| n \right\rangle$

The following commutators can be easily obtained by substituting the canonical commutation relation,

$[a,a^{\dagger}]=1,\qquad[N,a^{\dagger}]=a^{\dagger},\qquad[N,a]=-a,$

And the Hamilton operator can be expressed as

$H=\left(N%2B\frac{1}{2}\right)\hbar\omega,$

so the eigenstate of N is also the eigenstate of energy.

The commutation properties yields

$\begin{align} Na^{\dagger}|n\rangle&=\left(a^{\dagger}N%2B[N,a^{\dagger}]\right)|n\rangle\\&=\left(a^{\dagger}N%2Ba^{\dagger}\right)|n\rangle\\&=(n%2B1)a^{\dagger}|n\rangle, \end{align}$

and similarly,

$Na|n\rangle=(n-1)a|n\rangle.$

This means that a acts on $|n\rangle$ to produce, up to a multiplicative constant, $|n-1\rangle$ , and a^† acts on $|n\rangle$ to produce $|n%2B1\rangle$ . For this reason, a is called a "lowering operator", and a^† a "raising operator". The two operators together are called ladder operators. In quantum field theory, a and a^† are alternatively called "annihilation" and "creation" operators because they destroy and create particles, which correspond to our quanta of energy.

Given any energy eigenstate, we can act on it with the lowering operator, a, to produce another eigenstate with $\hbar \omega$ -less energy. By repeated application of the lowering operator, it seems that we can produce energy eigenstates down to E = −∞. However, since

$n=\langle n|N|n\rangle=\langle n|a^{\dagger}a|n\rangle=\left(a|n\rangle\right)^{\dagger}a|n\rangle\geqslant 0,$

the smallest eigen number is 0, and

$a \left| 0 \right\rangle = 0$ .

In this case, subsequent applications of the lowering operator will just produce zero kets, instead of additional energy eigenstates. Furthermore, we have shown above that

$H \left|0\right\rangle = \frac{\hbar\omega}{2} \left|0\right\rangle$

Finally, by acting on $\left| 0 \right \rangle$ with the raising operator and multiplying by suitable normalization factors, we can produce an infinite set of energy eigenstates $\left\{\left| 0 \right \rangle, \left| 1 \right \rangle, \left| 2 \right \rangle, ... , \left| n \right \rangle, ...\right\}$ , such that

$H \left|n\right\rangle = \hbar\omega \left(n %2B\frac{1}{2} \right) \left|n\right\rangle$

which matches the energy spectrum which we gave in the preceding section.

Arbitrary eigenstate can be expressed in terms of $| 0 \rangle$

$|n\rangle=\frac{\left(a^{\dagger}\right)^{n}}{\sqrt{n!}}|0\rangle.$

Proof:

$\begin{align} \langle n|aa^{\dagger}|n\rangle&=\langle n|\left([a,a^{\dagger}]%2Ba^{\dagger}a\right)|n\rangle=\langle n|\left(N%2B1\right)|n\rangle=n%2B1\\\Rightarrow a^{\dagger}|n\rangle&=\sqrt{n%2B1}|n%2B1\rangle\\\Rightarrow|n\rangle&=\frac{a^{\dagger}}{\sqrt{n}}|n-1\rangle=\frac{\left(a^{\dagger}\right)^{2}}{\sqrt{n(n-1)}}|n-2\rangle=\cdots=\frac{\left(a^{\dagger}\right)^{n}}{\sqrt{n!}}|0\rangle. \end{align}$

Natural length and energy scales

The quantum harmonic oscillator possesses natural scales for length and energy, which can be used to simplify the problem. These can be found by nondimensionalization. The result is that if we measure energy in units of $\hbar \omega$ and distance in units of $\left(\hbar / \left(m \omega\right)\right)^{1/2}$ , then the Schrödinger equation becomes:

$H = - {1\over2} {d^2 \over dx^2 } %2B {1 \over 2} x^2$ ,

and the energy eigenfunctions and eigenvalues become

$\left\langle x | \psi_n \right\rangle = {1 \over \sqrt{2^n n!}} \pi^{-1/4} \hbox{exp} (-x^2 / 2) H_n(x)$

$E_n = n %2B {1\over 2}$

where $H_n(x)$ are the Hermite polynomials.

To avoid confusion, we will not adopt these natural units in this article. However, they frequently come in handy when performing calculations.

Diatomic molecules

Main article: Diatomic molecule

The vibrations of a diatomic molecule are an example of a two-body version of the quantum harmonic oscillator. In this case, the angular frequency is given by

$\omega = \sqrt{\frac{k}{\mu}}$

where $\mu$ is the reduced mass and is determined by the mass of the two atoms. ^[4]

N-dimensional harmonic oscillator

The one-dimensional harmonic oscillator is readily generalizable to N dimensions, where N = 1, 2, 3, ... . In one dimension, the position of the particle was specified by a single coordinate, x. In N dimensions, this is replaced by N position coordinates, which we label x₁, ..., x_N. Corresponding to each position coordinate is a momentum; we label these p₁, ..., p_N. The canonical commutation relations between these operators are

$\begin{matrix} \left[x_i , p_j \right] &=& i\hbar\delta_{i,j} \\ \left[x_i , x_j \right] &=& 0 \\ \left[p_i , p_j \right] &=& 0 \end{matrix}$ .

The Hamiltonian for this system is

$H = \sum_{i=1}^N \left( {p_i^2 \over 2m} %2B {1\over 2} m \omega^2 x_i^2 \right)$ .

As the form of this Hamiltonian makes clear, the N-dimensional harmonic oscillator is exactly analogous to N independent one-dimensional harmonic oscillators with the same mass and spring constant. In this case, the quantities x₁, ..., x_N would refer to the positions of each of the N particles. This is a convenient property of the $r^2$ potential, which allows the potential energy to be separated into terms depending on one coordinate each.

This observation makes the solution straightforward. For a particular set of quantum numbers {n} the energy eigenfunctions for the N-dimensional oscillator are expressed in terms of the 1-dimensional eigenfunctions as:

$\langle \mathbf{x}|\psi_{\{n\}}\rangle =\prod_{i=1}^N\langle x_i|\psi_{n_i}\rangle$

In the ladder operator method, we define N sets of ladder operators,

$\begin{matrix} a_i &=& \sqrt{m\omega \over 2\hbar} \left(x_i %2B {i \over m \omega} p_i \right) \\ a^{\dagger}_i &=& \sqrt{m \omega \over 2\hbar} \left( x_i - {i \over m \omega} p_i \right) \end{matrix}$ .

By a procedure analogous to the one-dimensional case, we can then show that each of the a_i and a^†_i operators lower and raise the energy by ℏω respectively. The Hamiltonian is

$H = \hbar \omega \, \sum_{i=1}^N \left(a_i^\dagger \,a_i %2B \frac{1}{2}\right).$

This Hamiltonian is invariant under the dynamic symmetry group U(N) (the unitary group in N dimensions), defined by

$U\, a_i^\dagger \,U^\dagger = \sum_{j=1}^N a_j^\dagger\,U_{ji}\quad\hbox{for all}\quad U \in U(N),$

where $U_{ji}$ is an element in the defining matrix representation of U(N).

The energy levels of the system are

$E = \hbar \omega \left[(n_1 %2B \cdots %2B n_N) %2B {N\over 2}\right]$ .

$n_i = 0, 1, 2, \dots \quad (\hbox{the number of bosons in mode } i).$

As in the one-dimensional case, the energy is quantized. The ground state energy is N times the one-dimensional energy, as we would expect using the analogy to N independent one-dimensional oscillators. There is one further difference: in the one-dimensional case, each energy level corresponds to a unique quantum state. In N-dimensions, except for the ground state, the energy levels are degenerate, meaning there are several states with the same energy.

The degeneracy can be calculated relatively easily. As an example, consider the 3-dimensional case: Define n = n₁ + n₂ + n₃. All states with the same n will have the same energy. For a given n, we choose a particular n₁. Then n₂ + n₃ = n − n₁. There are n − n₁ + 1 possible groups {n₂, n₃}. n₂ can take on the values 0 to n − n₁, and for each n₂ the value of n₃ is fixed. The degree of degeneracy therefore is:

$g_n = \sum_{n_1=0}^n n - n_1 %2B 1 = \frac{(n%2B1)(n%2B2)}{2}$

Formula for general N and n [g_n being the dimension of the symmetric irreducible n^th power representation of the unitary group U(N)]:

$g_n = \binom{N%2Bn-1}{n}$

The special case N = 3, given above, follows directly from this general equation. This is however, only true for distinguishable particle, or one particle in N dimensions (as dimensions are distinguishable). For the case of N bosons in a one dimension harmonic trap, the degeneracy scales as the number of ways to partition an integer using integers less than or equal to N.

$g_n = p(N_{-},E)$

This arises due to the constraint of putting N quanta into a state ket where $\sum_{k=0}^\infty k n_k = E$ and $\sum_{k=0}^\infty n_k = N$ , which are the same constraints as in integer partition.

Example: 3D isotropic harmonic oscillator

The Schrödinger equation of a spherically-symmetric three-dimensional harmonic oscillator can be solved explicitly by separation of variables, see this article for the present case. This procedure is analogous to the separation performed in the hydrogen-like atom problem, but with the spherically symmetric potential

$V(r) = {1\over 2} \mu \omega^2 r^2,$

where $\mu$ is the mass of the problem. (Because m will be used below for the magnetic quantum number, mass is indicated by $\mu$ , instead of m, as earlier in this article.)

The solution reads

$\psi_{klm}(r,\theta,\phi) = N_{kl} r^{l}e^{-\nu r^2}{L_k}^{(l%2B{1\over 2})}(2\nu r^2) Y_{lm}(\theta,\phi)$

where

$N_{kl}=\sqrt{\sqrt{\frac{2\nu ^{3}}{\pi }}\frac{2^{k%2B2l%2B3}\;k!\;\nu ^{l}}{ (2k%2B2l%2B1)!!}}\,$ is a normalization constant.

$\nu \equiv {\mu \omega \over 2 \hbar}$

${L_k}^{(l%2B{1\over 2})}(2\nu r^2)$ are generalized Laguerre polynomials. The order k of the polynomial is a non-negative integer.

$Y_{lm}(\theta,\phi)\,$ is a spherical harmonic function.

$\hbar$ is the reduced Planck constant: $\hbar\equiv\frac{h}{2\pi}$ .

The energy eigenvalue is

$E=\hbar \omega \left(2k%2Bl%2B\frac{3}{2}\right)$

The energy is usually described by the single quantum number

$n\equiv 2k%2Bl\,$

Because k is a non-negative integer, for every even n we have $l=0,2...,n-2,n$ and for every odd n we have $l=1,3...,n-2,n$ . The magnetic quantum number m is an integer satisfying $-l \le m \le l$ , so for every n and l there are 2l+1 different quantum states, labeled by m. Thus, the degeneracy at level n is

$\sum_{l=\ldots,n-2,n} (2l%2B1) = {(n%2B1)(n%2B2)\over 2}$

where the sum starts from 0 or 1, according to whether n is even or odd. This result is in accordance with the dimension formula above.

Coupled harmonic oscillators

In this problem, we consider N equal masses which are connected to their neighbors by springs, in the limit of large N. The masses form a linear chain in one dimension, or a regular lattice in two or three dimensions.

As in the previous section, we denote the positions of the masses by x₁, x₂, ..., as measured from their equilibrium positions (i.e. x_k = 0 if particle k is at its equilibrium position.) In two or more dimensions, the xs are vector quantities. The Hamiltonian of the total system is

$H = \sum_{i=1}^N {p_i^2 \over 2m} %2B {1\over 2} m \omega^2\!\!\!\!\sum_{\{ij\} (nn)} \!\!\!(x_i - x_j)^2.$

The potential energy is summed over "nearest-neighbor" pairs, so there is one term for each spring.

Remarkably, there exists a coordinate transformation^[5] to turn this problem into a set of independent harmonic oscillators, each of which corresponds to a particular collective distortion of the lattice. These distortions display some particle-like properties, and are called phonons. Phonons occur in all (even amorphous) solids and are extremely important for understanding many of the phenomena studied in solid state physics.

References

^ Griffiths, David J. (2004). Introduction to Quantum Mechanics (2nd ed.). Prentice Hall. ISBN 0-13-805326-X.
^ Liboff, Richard L. (2002). Introductory Quantum Mechanics. Addison-Wesley. ISBN 0-8053-8714-5.
^ Rashid, Muneer A. (2006). "Transition amplitude for time-dependent linear harmonic oscillator with Linear time-dependent terms added to the Hamiltonian" (PDF-Microsoft PowerPoint). M.A. Rashid - Center for Advanced Mathematics and Physics. National Center for Physics. http://www.ncp.edu.pk/docs/12th_rgdocs/Munir-Rasheed.pdf. Retrieved 2010.
^ "Quantum Harmonic Oscillator". Hyperphysics. http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hosc.html. Retrieved 24 September 2009.
^ Armstrong; Zinner; Fedorov; Jensen (2011). "Analytic Harmonic Approach to the N-body problem". Journal of Physics B: Atomic, Molecular and Optical Physics 44 (5): 055303. arXiv:1011.2453v2. Bibcode 2011JPhB...44e5303A. doi:10.1088/0953-4075/44/5/055303.