Quantum harmonic oscillator

One-dimensional harmonic oscillator

Hamiltonian and energy eigenstates

The Hamiltonian of the particle is: $${\displaystyle \hat{H}=\frac{{\hat{p}}^2}{2m}+\frac{1}{2}k{\hat{x}}^2=\frac{{\hat{p}}^2}{2m}+\frac{1}{2}m{\omega }^2{\hat{x}}^2,}$$ where m is the particle's mass, k is the force constant, $\omega =\sqrt{k/m}$ is the angular frequency of the oscillator, ${\displaystyle \hat{x}}$ is the position operator (given by x in the coordinate basis), and ${\displaystyle \hat{p}}$ is the momentum operator (given by ${\displaystyle \hat{p}=-i\hslash \mathrm{\partial }/\mathrm{\partial }x}$ in the coordinate basis). The first term in the Hamiltonian represents the kinetic energy of the particle, and the second term represents its potential energy, as in Hooke's law.

The time-independent Schrödinger equation (TISE) is, $${\displaystyle \hat{H}|\psi ⟩=E|\psi ⟩,}$$ where ${\displaystyle E}$ denotes a real number (which needs to be determined) that will specify a time-independent energy level, or eigenvalue, and the solution ${\displaystyle |\psi ⟩}$ denotes that level's energy eigenstate.

Then solve the differential equation representing this eigenvalue problem in the coordinate basis, for the wave function ${\displaystyle ⟨x|\psi ⟩=\psi (x)}$, using a spectral method. It turns out that there is a family of solutions. In this basis, they amount to Hermite functions, $${\displaystyle {\psi }_n(x)=\frac{1}{\sqrt{2^{n}n!}}{\left(\frac{m\omega }{\pi \hslash }\right)}^{1/4}{e}^{-\frac{m\omega x^2}{2\hslash }}{H}_n\left(\sqrt{\frac{m\omega }{\hslash }}x\right),n=0,1,2,\dots .}$$

The functions ${\displaystyle H_n}$ are the physicists' Hermite polynomials, $${\displaystyle H_n(z)=(-1{)}^n{e}^{z^2}\frac{d^n}{d{z}^n}\left(e^{-z^2}\right).}$$

The corresponding energy levels are $${\displaystyle E_n=\hslash \omega (n+\frac{1}{2}).}$$The expectation values of position and momentum combined with variance of each variable can be derived from the wavefunction to understand the behavior of the energy eigenkets. They are shown to be $⟨\hat{x}⟩=0$ and $⟨\hat{p}⟩=0$ owing to the symmetry of the problem, whereas:

$${\displaystyle \begin{array}{cc}⟨x^2⟩& =(2n+1)\frac{\hslash }{2m\omega }={\sigma }_x^2\\ ⟨p^2⟩& =(2n+1)\frac{m\hslash \omega }{2}={\sigma }_p^2\end{array}}$$

The variance in both position and momentum are observed to increase for higher energy levels. The lowest energy level has value of ${\sigma }_x{\sigma }_p=\frac{\hslash }{2}$ which is its minimum value due to uncertainty relation and also corresponds to a Gaussian wavefunction.

This energy spectrum is noteworthy for four reasons. First, the energies are quantized, meaning that only discrete energy values (integer-plus-half multiples of ħω) 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 ħω/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. Fourth, the energy levels are nondegenerate implying that every eigenvalue is associated with only one solution (state).

The ground state probability density is concentrated at the origin, which means the particle spends most of its time at the bottom of the potential well, as one would expect for a state with little energy. As the energy increases, the probability density peaks at the classical "turning points", where the state's energy coincides with the potential energy. (See the discussion below of the highly excited states.) This is consistent with the classical harmonic oscillator, in which the particle spends more of its time (and is therefore more likely to be found) near the turning points, where it is moving the slowest. The correspondence principle is thus satisfied. Moreover, special nondispersive wave packets, with minimum uncertainty, called coherent states oscillate very much like classical objects, as illustrated in the figure; they are not eigenstates of the Hamiltonian.

Ladder operator method

The "ladder operator" method, developed by Paul Dirac, allows extraction of the energy eigenvalues without directly solving the differential equation. It is generalizable to more complicated problems, notably in quantum field theory. Following this approach, we define the operators ${\displaystyle \hat{a}}$ and its adjoint ${\displaystyle {\hat{a}}^{†}}$, $${\displaystyle \begin{array}{cc}\hat{a}& =\sqrt{\frac{m\omega }{2\hslash }}\left(\hat{x}+\frac{i}{m\omega }\hat{p}\right)\\ {\hat{a}}^{†}& =\sqrt{\frac{m\omega }{2\hslash }}\left(\hat{x}-\frac{i}{m\omega }\hat{p}\right)\end{array}}$$Note these operators classically are exactly the generators of normalized rotation in the phase space of ${\displaystyle x}$ and ${\displaystyle m\frac{dx}{dt}}$, i.e they describe the forwards and backwards evolution in time of a classical harmonic oscillator.

These operators lead to the following representation of ${\displaystyle \hat{x}}$ and ${\displaystyle \hat{p}}$, $${\displaystyle \begin{array}{cc}\hat{x}& =\sqrt{\frac{\hslash }{2m\omega }}({\hat{a}}^{†}+\hat{a})\\ \hat{p}& =i\sqrt{\frac{\hslash m\omega }{2}}({\hat{a}}^{†}-\hat{a}).\end{array}}$$

The operator a is not Hermitian, since itself and its adjoint a^† are not equal. The energy eigenstates |n⟩, when operated on by these ladder operators, give $${\displaystyle \begin{array}{cc}{\hat{a}}^{†}|n⟩& =\sqrt{n+1}|n+1⟩\\ \hat{a}|n⟩& =\sqrt{n}|n-1⟩.\end{array}}$$

From the relations above, we can also define a number operator N, which has the following property: $${\displaystyle \begin{array}{cc}\hat{N}& ={\hat{a}}^{†}\hat{a}\\ \hat{N}|n⟩& =n|n⟩.\end{array}}$$

The following commutators can be easily obtained by substituting the canonical commutation relation, $${\displaystyle [\hat{a},{\hat{a}}^{†}]=1,[\hat{N},{\hat{a}}^{†}]={\hat{a}}^{†},[\hat{N},\hat{a}]=-\hat{a},}$$

and the Hamilton operator can be expressed as $${\displaystyle \hat{H}=\hslash \omega \left(\hat{N}+\frac{1}{2}\right),}$$

so the eigenstates of ${\displaystyle \hat{N}}$ are also the eigenstates of energy. To see that, we can apply ${\displaystyle \hat{H}}$ to a number state ${\displaystyle |n⟩}$:

$${\displaystyle \hat{H}|n⟩=\hslash \omega \left(\hat{N}+\frac{1}{2}\right)|n⟩.}$$

Using the property of the number operator ${\displaystyle \hat{N}}$:

$${\displaystyle \hat{N}|n⟩=n|n⟩,}$$

we get:

$${\displaystyle \hat{H}|n⟩=\hslash \omega \left(n+\frac{1}{2}\right)|n⟩.}$$

Thus, since ${\displaystyle |n⟩}$ solves the TISE for the Hamiltonian operator ${\displaystyle \hat{H}}$, is also one of its eigenstates with the corresponding eigenvalue:

$${\displaystyle E_n=\hslash \omega \left(n+\frac{1}{2}\right).}$$

QED.

The commutation property yields $${\displaystyle \begin{array}{cc}\hat{N}{\hat{a}}^{†}|n⟩& =\left({\hat{a}}^{†}\hat{N}+[\hat{N},{\hat{a}}^{†}]\right)|n⟩\\ & =\left({\hat{a}}^{†}\hat{N}+{\hat{a}}^{†}\right)|n⟩\\ & =(n+1){\hat{a}}^{†}|n⟩,\end{array}}$$

and similarly, $${\displaystyle \hat{N}\hat{a}|n⟩=(n-1)\hat{a}|n⟩.}$$

This means that ${\displaystyle \hat{a}}$ acts on ${\displaystyle |n⟩}$ to produce, up to a multiplicative constant, ${\displaystyle |n-1⟩}$, and ${\displaystyle {\hat{a}}^{†}}$ acts on ${\displaystyle |n⟩}$ to produce ${\displaystyle |n+1⟩}$. For this reason, ${\displaystyle \hat{a}}$ is called an annihilation operator ("lowering operator"), and ${\displaystyle {\hat{a}}^{†}}$ a creation operator ("raising operator"). The two operators together are called ladder operators.

Given any energy eigenstate, we can act on it with the lowering operator, a, to produce another eigenstate with ħω less energy. By repeated application of the lowering operator, it seems that we can produce energy eigenstates down to E = −∞. However, since $${\displaystyle n=⟨n|N|n⟩=⟨n|a^{†}a|n⟩=(a|n⟩{)}^{†}a|n⟩⩾0,}$$

the smallest eigenvalue of the number operator is 0, and $${\displaystyle a|0⟩=0.}$$

In this case, subsequent applications of the lowering operator will just produce zero, instead of additional energy eigenstates. Furthermore, we have shown above that $${\displaystyle \hat{H}|0⟩=\frac{\hslash \omega }{2}|0⟩}$$

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

such that $${\displaystyle \hat{H}|n⟩=\hslash \omega \left(n+\frac{1}{2}\right)|n⟩,}$$ which matches the energy spectrum given in the preceding section.

Arbitrary eigenstates can be expressed in terms of ${\displaystyle |0⟩}$, $${\displaystyle |n⟩=\frac{(a^{†}{)}^n}{\sqrt{n!}}|0⟩.}$$

Proof

$${\displaystyle \begin{array}{cc}⟨n|a{a}^{†}|n⟩& =⟨n|\left([a,a^{†}]+a^{†}a\right)|n⟩=⟨n|\left(N+1\right)|n⟩=n+1\\ \Rightarrow a^{†}|n⟩& =\sqrt{n+1}|n+1⟩\\ \Rightarrow |n⟩& =\frac{1}{\sqrt{n}}{a}^{†}|n-1⟩=\frac{1}{\sqrt{n(n-1)}}{\left(a^{†}\right)}^2|n-2⟩=\cdots =\frac{1}{\sqrt{n!}}{\left(a^{†}\right)}^n|0⟩.\end{array}}$$

Analytical questions

The preceding analysis is algebraic, using only the commutation relations between the raising and lowering operators. Once the algebraic analysis is complete, one should turn to analytical questions. First, one should find the ground state, that is, the solution of the equation ${\displaystyle a{\psi }_0=0}$. In the position representation, this is the first-order differential equation $${\displaystyle \left(x+\frac{\hslash }{m\omega }\frac{d}{dx}\right){\psi }_0=0,}$$ whose solution is easily found to be the Gaussian $${\displaystyle {\psi }_0(x)=C{e}^{-\frac{m\omega x^2}{2\hslash }}.}$$ Conceptually, it is important that there is only one solution of this equation; if there were, say, two linearly independent ground states, we would get two independent chains of eigenvectors for the harmonic oscillator. Once the ground state is computed, one can show inductively that the excited states are Hermite polynomials times the Gaussian ground state, using the explicit form of the raising operator in the position representation. One can also prove that, as expected from the uniqueness of the ground state, the Hermite functions energy eigenstates ${\displaystyle {\psi }_n}$ constructed by the ladder method form a complete orthonormal set of functions.

Given that Hermite functions are either even or odd, it can be shown that the average displacement and average momentum is 0 for all states in QHO.

Explicitly connecting with the previous section, the ground state |0⟩ in the position representation is determined by ${\displaystyle a|0⟩=0}$, $${\displaystyle ⟨x\mid a\mid 0⟩=0\Rightarrow \left(x+\frac{\hslash }{m\omega }\frac{d}{dx}\right)⟨x\mid 0⟩=0\Rightarrow }$$ $${\displaystyle ⟨x\mid 0⟩={\left(\frac{m\omega }{\pi \hslash }\right)}^{\frac{1}{4}}\exp \left(-\frac{m\omega }{2\hslash }{x}^2\right)={\psi }_0,}$$ hence $${\displaystyle ⟨x\mid a^{†}\mid 0⟩={\psi }_1(x),}$$ so that ${\displaystyle {\psi }_1(x,t)=⟨x\mid e^{-3i\omega t/2}{a}^{†}\mid 0⟩}$, and so on.

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 energy is measured in units of ħω and distance in units of √ħ/(mω), then the Hamiltonian simplifies to $${\displaystyle H=-\frac{1}{2}\frac{d^2}{d{x}^2}+\frac{1}{2}{x}^2,}$$ while the energy eigenfunctions and eigenvalues simplify to Hermite functions and integers offset by a half, $${\displaystyle {\psi }_n(x)=⟨x\mid n⟩=\frac{1}{\sqrt{2^{n}n!}}{\pi }^{-1/4}\exp \left(-x^2/2\right)H_n(x),}$$ $${\displaystyle E_n=n+\frac{1}{2},}$$ where H_n(x) are the Hermite polynomials.

To avoid confusion, these "natural units" will mostly not be adopted in this article. However, they frequently come in handy when performing calculations, by bypassing clutter.

For example, the fundamental solution (propagator) of H − i∂_t, the time-dependent Schrödinger operator for this oscillator, simply boils down to the Mehler kernel, $${\displaystyle ⟨x|\exp (-itH)|y⟩\equiv K(x,y;t)=\frac{1}{\sqrt{2\pi i\sin t}}\exp \left(\frac{i}{2\sin t}\left(\left(x^2+y^2\right)\cos t-2xy\right)\right),}$$ where K(x,y;0) = δ(x − y). The most general solution for a given initial configuration ψ(x,0) then is simply $${\displaystyle \psi (x,t)=\int dyK(x,y;t)\psi (y,0).}$$

Coherent states

The coherent states (also known as Glauber states) of the harmonic oscillator are special nondispersive wave packets, with minimum uncertainty σ_x σ_p = ℏ⁄2, whose observables' expectation values evolve like a classical system. They are eigenvectors of the annihilation operator, not the Hamiltonian, and form an overcomplete basis which consequentially lacks orthogonality.

The coherent states are indexed by ${\displaystyle \alpha \in \mathbb{C}}$ and expressed in the |n⟩ basis as

$${\displaystyle \begin{array}{cc}|\alpha ⟩& =\sum _{n=0}^{\mathrm{\infty }}|n⟩⟨n|\alpha ⟩\\ & =e^{-\frac{1}{2}|\alpha {|}^2}\sum _{n=0}^{\mathrm{\infty }}\frac{{\alpha }^n}{\sqrt{n!}}|n⟩\\ & =e^{-\frac{1}{2}|\alpha {|}^2}{e}^{\alpha a^{†}}{e}^{-{\alpha }^{\ast }a}|0⟩.\end{array}}$$

Since coherent states are not energy eigenstates, their time evolution is not a simple shift in wavefunction phase. The time-evolved states are, however, also coherent states but with phase-shifting parameter α instead: ${\displaystyle \alpha (t)=\alpha (0)e^{-i\omega t}={\alpha }_0{e}^{-i\omega t}}$.

$${\displaystyle \begin{array}{cc}|\alpha (t)⟩& =\sum _{n=0}^{\mathrm{\infty }}{e}^{-i\left(n+\frac{1}{2}\right)\omega t}|n⟩⟨n|\alpha ⟩\\ & =e^{\frac{-i\omega t}{2}}{e}^{-\frac{1}{2}|\alpha {|}^2}\sum _{n=0}^{\mathrm{\infty }}\frac{(\alpha e^{-i\omega t}{)}^n}{\sqrt{n!}}|n⟩\\ & =e^{-\frac{i\omega t}{2}}|\alpha e^{-i\omega t}⟩\end{array}}$$

Because ${\displaystyle a|0⟩=0}$ and via the Kermack-McCrae identity, the last form is equivalent to a unitary displacement operator acting on the ground state: ${\displaystyle |\alpha ⟩=e^{\alpha {\hat{a}}^{†}-{\alpha }^{\ast }\hat{a}}|0⟩=D(\alpha )|0⟩}$. Calculating the expectation values:

$${\displaystyle \begin{array}{cc}⟨\hat{x}{⟩}_{\alpha (t)}& =\sqrt{\frac{2\hslash }{m\omega }}\left|{\alpha }_0\right|\cos (\omega t-\varphi )\\ ⟨\hat{p}{⟩}_{\alpha (t)}& =-\sqrt{2m\hslash \omega }\left|{\alpha }_0\right|\sin (\omega t-\varphi )\end{array}}$$

where ${\displaystyle \varphi }$ is the phase contributed by complex α. These equations confirm the oscillating behavior of the particle.

The uncertainties calculated using the numeric method are:

$${\displaystyle \begin{array}{cc}{\sigma }_x(t)& =\sqrt{\frac{\hslash }{2m\omega }}\\ {\sigma }_p(t)& =\sqrt{\frac{m\hslash \omega }{2}}\end{array}}$$

which gives ${\sigma }_x(t){\sigma }_p(t)=\frac{\hslash }{2}$. Since the only wavefunction that can have lowest position–momentum uncertainty, $\frac{\hslash }{2}$, is a Gaussian wavefunction, and since the coherent state wavefunction has minimum position–momentum uncertainty, we note that the general Gaussian wavefunction in quantum mechanics has the form:$${\displaystyle {\psi }_{\alpha }(x^{\prime })={\left(\frac{m\omega }{\pi \hslash }\right)}^{\frac{1}{4}}{e}^{\frac{i}{\hslash }⟨\hat{p}{⟩}_{\alpha }(x^{\prime }-\frac{⟨\hat{x}{⟩}_{\alpha }}{2})-\frac{m\omega }{2\hslash }(x^{\prime }-⟨\hat{x}{⟩}_{\alpha }{)}^2}.}$$Substituting the expectation values as a function of time, gives the required time varying wavefunction.

The probability of each energy eigenstates can be calculated to find the energy distribution of the wavefunction:

$${\displaystyle P(E_n)={\left|⟨n|\alpha ⟩\right|}^2=\frac{e^{-|\alpha {|}^2}|\alpha {|}^{2n}}{n!}}$$

which corresponds to a Poisson distribution.

Highly excited states

When n is large, the eigenstates are localized into the classical allowed region, that is, the region in which a classical particle with energy E_n can move. The eigenstates are peaked near the turning points: the points at the ends of the classically allowed region where the classical particle changes direction. This phenomenon can be verified through asymptotics of the Hermite polynomials, and also through the WKB approximation.

The frequency of oscillation at x is proportional to the momentum p(x) of a classical particle of energy E_n and position x. Furthermore, the square of the amplitude (determining the probability density) is inversely proportional to p(x), reflecting the length of time the classical particle spends near x. The system behavior in a small neighborhood of the turning point does not have a simple classical explanation, but can be modeled using an Airy function. Using properties of the Airy function, one may estimate the probability of finding the particle outside the classically allowed region, to be approximately $${\displaystyle \frac{2}{n^{1/3}{3}^{2/3}{\mathrm{\Gamma }}^2(\frac{1}{3})}=\frac{1}{n^{1/3}\cdot \mathrm{7.46408092658...}}}$$ This is also given, asymptotically, by the integral $${\displaystyle \frac{1}{2\pi }{\int }_0^{\mathrm{\infty }}{e}^{(2n+1)\left(x-\frac{1}{2}\sinh (2x)\right)}dx.}$$

Phase space solutions

In the phase space formulation of quantum mechanics, eigenstates of the quantum harmonic oscillator in several different representations of the quasiprobability distribution can be written in closed form. The most widely used of these is for the Wigner quasiprobability distribution.

The Wigner quasiprobability distribution for the energy eigenstate |n⟩ is, in the natural units described above, $${\displaystyle F_n(x,p)=\frac{(-1{)}^n}{\pi \hslash }{L}_n\left(2(x^2+p^2)\right)e^{-(x^2+p^2)},}$$ where L_n are the Laguerre polynomials. This example illustrates how the Hermite and Laguerre polynomials are linked through the Wigner map.

Meanwhile, the Husimi Q function of the harmonic oscillator eigenstates have an even simpler form. If we work in the natural units described above, we have $${\displaystyle Q_n(x,p)=\frac{(x^2+p^2{)}^n}{n!}\frac{e^{-(x^2+p^2)}}{\pi }}$$ This claim can be verified using the Segal–Bargmann transform. Specifically, since the raising operator in the Segal–Bargmann representation is simply multiplication by ${\displaystyle z=x+ip}$ and the ground state is the constant function 1, the normalized harmonic oscillator states in this representation are simply ${\displaystyle z^n/\sqrt{n!}}$ . At this point, we can appeal to the formula for the Husimi Q function in terms of the Segal–Bargmann transform.

Two-dimensional harmonic oscillators

The two-dimensional Cartesian harmonic oscillator and the two-dimensional isotropic harmonic oscillator in cylindrical coordinates have been treated in detail in the book of Müller-Kirsten.