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Quantum harmonic oscillator

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(Redirected fromHarmonic potential)

Important, well-understood quantum mechanical model
Part of a series of articles about
Quantum mechanics
iℏddt|Ψ⟩=H^|Ψ⟩{\displaystyle i\hbar {\frac {d}{dt}}|\Psi \rangle ={\hat {H}}|\Psi \rangle }
"QHO" redirects here. It is also theIATA airport code forall airports in the Houston area.
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Some trajectories of aharmonic oscillator according toNewton's laws ofclassical mechanics (A–B), and according to theSchrödinger equation ofquantum mechanics (C–H). In A–B, the particle (represented as a ball attached to aspring) oscillates back and forth. In C–H, some solutions to the Schrödinger equation are shown, where the horizontal axis is position, and the vertical axis is the real part (blue) or imaginary part (red) of thewavefunction. C, D, E, F, but not G, H, areenergy eigenstates. H is acoherent state—a quantum state that approximates the classical trajectory.

Thequantum harmonic oscillator is thequantum-mechanical analog of theclassical harmonic oscillator. Because an arbitrary smoothpotential can usually be approximated as aharmonic potential at the vicinity of a stableequilibrium 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][4]

One-dimensional harmonic oscillator

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Hamiltonian and energy eigenstates

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Wavefunction representations for the first eight bound eigenstates,n = 0 to 7. The horizontal axis shows the positionx.
Corresponding probability densities.

TheHamiltonian of the particle is:H^=p^22m+12kx^2=p^22m+12mω2x^2,{\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}\,,}wherem is the particle's mass,k is the force constant,ω=k/m{\textstyle \omega ={\sqrt {k/m}}} is theangular frequency of the oscillator,x^{\displaystyle {\hat {x}}} is theposition operator (given byx in the coordinate basis), andp^{\displaystyle {\hat {p}}} is themomentum operator (given byp^=−iℏ∂/∂x{\displaystyle {\hat {p}}=-i\hbar \,\partial /\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 inHooke's law.[5]

The time-independentSchrödinger equation (TISE) is,H^|ψ⟩=E|ψ⟩ ,{\displaystyle {\hat {H}}\left|\psi \right\rangle =E\left|\psi \right\rangle ~,}whereE{\displaystyle E} denotes a real number (which needs to be determined) that will specify a time-independentenergy level, oreigenvalue, and the solution|ψ⟩{\displaystyle |\psi \rangle } denotes that level's energyeigenstate.[6]

Then solve the differential equation representing this eigenvalue problem in the coordinate basis, for thewave function⟨x|ψ⟩=ψ(x){\displaystyle \langle x|\psi \rangle =\psi (x)}, using aspectral method. It turns out that there is a family of solutions. In this basis, they amount to Hermite functions,[7][8]ψn(x)=12nn!(mωπℏ)1/4e−mωx22ℏHn(mωℏx),n=0,1,2,….{\displaystyle \psi _{n}(x)={\frac {1}{\sqrt {2^{n}\,n!}}}\left({\frac {m\omega }{\pi \hbar }}\right)^{1/4}e^{-{\frac {m\omega x^{2}}{2\hbar }}}H_{n}\left({\sqrt {\frac {m\omega }{\hbar }}}x\right),\qquad n=0,1,2,\ldots .}

The functionsHn{\displaystyle H_{n}} are the physicists'Hermite polynomials,Hn(z)=(−1)n ez2dndzn(e−z2).{\displaystyle H_{n}(z)=(-1)^{n}~e^{z^{2}}{\frac {d^{n}}{dz^{n}}}\left(e^{-z^{2}}\right).}

The corresponding energy levels are[9]En=ℏω(n+12).{\displaystyle E_{n}=\hbar \omega {\bigl (}n+{\tfrac {1}{2}}{\bigr )}.}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⟨x^⟩=0{\textstyle \langle {\hat {x}}\rangle =0} and⟨p^⟩=0{\textstyle \langle {\hat {p}}\rangle =0} owing to the symmetry of the problem, whereas:

⟨x^2⟩=(2n+1)ℏ2mω=σx2{\displaystyle \langle {\hat {x}}^{2}\rangle =(2n+1){\frac {\hbar }{2m\omega }}=\sigma _{x}^{2}}

⟨p^2⟩=(2n+1)mℏω2=σp2{\displaystyle \langle {\hat {p}}^{2}\rangle =(2n+1){\frac {m\hbar \omega }{2}}=\sigma _{p}^{2}}

The variance in both position and momentum are observed to increase for higher energy levels. The lowest energy level has value ofσxσp=ℏ2{\textstyle \sigma _{x}\sigma _{p}={\frac {\hbar }{2}}} which is its minimum value due to uncertainty relation and also corresponds to a gaussian wavefunction.[10]

This energy spectrum is noteworthy for four reasons. First, the energies are quantized, meaning that only discrete energy values (integer-plus-half multiples ofħω) 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 theBohr model of the atom, or theparticle in a box. Third, the lowest achievable energy (the energy of then = 0 state, called theground state) is not equal to the minimum of the potential well, butħω/2 above it; this is calledzero-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 theHeisenberg uncertainty principle. Fourth, the energy levels are nondegenerate implying that every eigenvalue is associated with only one solution (state).[11]

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. Thecorrespondence principle is thus satisfied. Moreover, special nondispersivewave packets, with minimum uncertainty, calledcoherent states oscillate very much like classical objects, as illustrated in the figure; they arenot eigenstates of the Hamiltonian.

Ladder operator method

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Probability densities |ψn(x)|2 for the bound eigenstates, beginning with the ground state (n = 0) at the bottom and increasing in energy toward the top. The horizontal axis shows the positionx, and brighter colors represent higher probability densities.

The "ladder operator" method, developed byPaul Dirac, allows extraction of the energy eigenvalues without directly solving the differential equation.[12] It is generalizable to more complicated problems, notably inquantum field theory. Following this approach, we define the operatorsa^{\displaystyle {\hat {a}}} and itsadjointa^†{\displaystyle {\hat {a}}^{\dagger }},a^=mω2ℏ(x^+imωp^)a^†=mω2ℏ(x^−imωp^){\displaystyle {\begin{aligned}{\hat {a}}&={\sqrt {m\omega \over 2\hbar }}\left({\hat {x}}+{i \over m\omega }{\hat {p}}\right)\\{\hat {a}}^{\dagger }&={\sqrt {m\omega \over 2\hbar }}\left({\hat {x}}-{i \over m\omega }{\hat {p}}\right)\end{aligned}}}Note these operators classically are exactly thegenerators of normalized rotation in the phase space ofx{\displaystyle x} andmdxdt{\displaystyle m{\frac {dx}{dt}}},i.e they describe the forwards and backwards evolution in time of a classical harmonic oscillator.[clarification needed]

These operators lead to the following representation ofx^{\displaystyle {\hat {x}}} andp^{\displaystyle {\hat {p}}},x^=ℏ2mω(a^†+a^)p^=iℏmω2(a^†−a^) .{\displaystyle {\begin{aligned}{\hat {x}}&={\sqrt {\frac {\hbar }{2m\omega }}}({\hat {a}}^{\dagger }+{\hat {a}})\\{\hat {p}}&=i{\sqrt {\frac {\hbar m\omega }{2}}}({\hat {a}}^{\dagger }-{\hat {a}})~.\end{aligned}}}

The operatora is notHermitian, since itself and its adjointa† are not equal. The energy eigenstates|n⟩, when operated on by these ladder operators, givea^†|n⟩=n+1|n+1⟩a^|n⟩=n|n−1⟩.{\displaystyle {\begin{aligned}{\hat {a}}^{\dagger }|n\rangle &={\sqrt {n+1}}|n+1\rangle \\{\hat {a}}|n\rangle &={\sqrt {n}}|n-1\rangle .\end{aligned}}}

From the relations above, we can also define a number operatorN, which has the following property:N^=a^†a^N^|n⟩=n|n⟩.{\displaystyle {\begin{aligned}{\hat {N}}&={\hat {a}}^{\dagger }{\hat {a}}\\{\hat {N}}\left|n\right\rangle &=n\left|n\right\rangle .\end{aligned}}}

The followingcommutators can be easily obtained by substituting thecanonical commutation relation,[a^,a^†]=1,[N^,a^†]=a^†,[N^,a^]=−a^,{\displaystyle [{\hat {a}},{\hat {a}}^{\dagger }]=1,\qquad [{\hat {N}},{\hat {a}}^{\dagger }]={\hat {a}}^{\dagger },\qquad [{\hat {N}},{\hat {a}}]=-{\hat {a}},}

and the Hamilton operator can be expressed asH^=ℏω(N^+12),{\displaystyle {\hat {H}}=\hbar \omega \left({\hat {N}}+{\frac {1}{2}}\right),}

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

H^|n⟩=ℏω(N^+12)|n⟩.{\displaystyle {\hat {H}}|n\rangle =\hbar \omega \left({\hat {N}}+{\frac {1}{2}}\right)|n\rangle .}

Using the property of the number operatorN^{\displaystyle {\hat {N}}}:

N^|n⟩=n|n⟩,{\displaystyle {\hat {N}}|n\rangle =n|n\rangle ,}

we get:

H^|n⟩=ℏω(n+12)|n⟩.{\displaystyle {\hat {H}}|n\rangle =\hbar \omega \left(n+{\frac {1}{2}}\right)|n\rangle .}

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

En=ℏω(n+12).{\displaystyle E_{n}=\hbar \omega \left(n+{\frac {1}{2}}\right).}

QED.

The commutation property yieldsN^a^†|n⟩=(a^†N^+[N^,a^†])|n⟩=(a^†N^+a^†)|n⟩=(n+1)a^†|n⟩,{\displaystyle {\begin{aligned}{\hat {N}}{\hat {a}}^{\dagger }|n\rangle &=\left({\hat {a}}^{\dagger }{\hat {N}}+[{\hat {N}},{\hat {a}}^{\dagger }]\right)|n\rangle \\&=\left({\hat {a}}^{\dagger }{\hat {N}}+{\hat {a}}^{\dagger }\right)|n\rangle \\&=(n+1){\hat {a}}^{\dagger }|n\rangle ,\end{aligned}}}

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

This means thata^{\displaystyle {\hat {a}}} acts on|n⟩{\displaystyle |n\rangle } to produce, up to a multiplicative constant,|n−1⟩{\displaystyle |n-1\rangle }, anda^†{\displaystyle {\hat {a}}^{\dagger }} acts on|n⟩{\displaystyle |n\rangle } to produce|n+1⟩{\displaystyle |n+1\rangle }. For this reason,a^{\displaystyle {\hat {a}}} is called anannihilation operator ("lowering operator"), anda^†{\displaystyle {\hat {a}}^{\dagger }} acreation operator ("raising operator"). The two operators together are calledladder operators.

Given any energy eigenstate, we can act on it with the lowering operator,a, to produce another eigenstate withħω less energy. By repeated application of the lowering operator, it seems that we can produce energy eigenstates down toE = −∞. However, sincen=⟨n|N|n⟩=⟨n|a†a|n⟩=(a|n⟩)†a|n⟩⩾0,{\displaystyle n=\langle n|N|n\rangle =\langle n|a^{\dagger }a|n\rangle ={\Bigl (}a|n\rangle {\Bigr )}^{\dagger }a|n\rangle \geqslant 0,}

the smallest eigenvalue of the number operator is 0, anda|0⟩=0.{\displaystyle a\left|0\right\rangle =0.}

In this case, subsequent applications of the lowering operator will just produce zero, instead of additional energy eigenstates. Furthermore, we have shown above thatH^|0⟩=ℏω2|0⟩{\displaystyle {\hat {H}}\left|0\right\rangle ={\frac {\hbar \omega }{2}}\left|0\right\rangle }

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

such thatH^|n⟩=ℏω(n+12)|n⟩,{\displaystyle {\hat {H}}\left|n\right\rangle =\hbar \omega \left(n+{\frac {1}{2}}\right)\left|n\right\rangle ,}which matches the energy spectrum given in the preceding section.

Arbitrary eigenstates can be expressed in terms of|0⟩{\displaystyle |0\rangle },[13]|n⟩=(a†)nn!|0⟩.{\displaystyle |n\rangle ={\frac {(a^{\dagger })^{n}}{\sqrt {n!}}}|0\rangle .}

Proof

⟨n|aa†|n⟩=⟨n|([a,a†]+a†a)|n⟩=⟨n|(N+1)|n⟩=n+1⇒a†|n⟩=n+1|n+1⟩⇒|n⟩=1na†|n−1⟩=1n(n−1)(a†)2|n−2⟩=⋯=1n!(a†)n|0⟩.{\displaystyle {\begin{aligned}\langle n|aa^{\dagger }|n\rangle &=\langle n|\left([a,a^{\dagger }]+a^{\dagger }a\right)\left|n\right\rangle =\langle n|\left(N+1\right)|n\rangle =n+1\\[1ex]\Rightarrow a^{\dagger }|n\rangle &={\sqrt {n+1}}|n+1\rangle \\[1ex]\Rightarrow |n\rangle &={\frac {1}{\sqrt {n}}}a^{\dagger }\left|n-1\right\rangle ={\frac {1}{\sqrt {n(n-1)}}}\left(a^{\dagger }\right)^{2}\left|n-2\right\rangle =\cdots ={\frac {1}{\sqrt {n!}}}\left(a^{\dagger }\right)^{n}\left|0\right\rangle .\end{aligned}}}

Analytical questions

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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 equationaψ0=0{\displaystyle a\psi _{0}=0}. In the position representation, this is the first-order differential equation(x+ℏmωddx)ψ0=0,{\displaystyle \left(x+{\frac {\hbar }{m\omega }}{\frac {d}{dx}}\right)\psi _{0}=0,}whose solution is easily found to be theGaussian[nb 1]ψ0(x)=Ce−mωx22ℏ.{\displaystyle \psi _{0}(x)=Ce^{-{\frac {m\omega x^{2}}{2\hbar }}}.}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 eigenstatesψn{\displaystyle \psi _{n}} constructed by the ladder method form acomplete orthonormal set of functions.[14]

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.[11]

Explicitly connecting with the previous section, the ground state |0⟩ in the position representation is determined bya|0⟩=0{\displaystyle a|0\rangle =0},⟨x∣a∣0⟩=0⇒(x+ℏmωddx)⟨x∣0⟩=0⇒{\displaystyle \left\langle x\mid a\mid 0\right\rangle =0\qquad \Rightarrow \left(x+{\frac {\hbar }{m\omega }}{\frac {d}{dx}}\right)\left\langle x\mid 0\right\rangle =0\qquad \Rightarrow }⟨x∣0⟩=(mωπℏ)14exp⁡(−mω2ℏx2)=ψ0 ,{\displaystyle \left\langle x\mid 0\right\rangle =\left({\frac {m\omega }{\pi \hbar }}\right)^{\frac {1}{4}}\exp \left(-{\frac {m\omega }{2\hbar }}x^{2}\right)=\psi _{0}~,}hence⟨x∣a†∣0⟩=ψ1(x) ,{\displaystyle \langle x\mid a^{\dagger }\mid 0\rangle =\psi _{1}(x)~,}so thatψ1(x,t)=⟨x∣e−3iωt/2a†∣0⟩{\displaystyle \psi _{1}(x,t)=\langle x\mid e^{-3i\omega t/2}a^{\dagger }\mid 0\rangle }, and so on.

Natural length and energy scales

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See also:Path integral formulation § Simple harmonic oscillator

The quantum harmonic oscillator possesses natural scales for length and energy, which can be used to simplify the problem. These can be found bynondimensionalization.

The result is that, ifenergy is measured in units ofħω anddistance in units of√ħ/(mω), then the Hamiltonian simplifies toH=−12d2dx2+12x2,{\displaystyle H=-{\frac {1}{2}}{d^{2} \over dx^{2}}+{\frac {1}{2}}x^{2},}while the energy eigenfunctions and eigenvalues simplify to Hermite functions and integers offset by a half,ψn(x)=⟨x∣n⟩=12nn! π−1/4exp⁡(−x2/2) Hn(x),{\displaystyle \psi _{n}(x)=\left\langle x\mid n\right\rangle ={1 \over {\sqrt {2^{n}n!}}}~\pi ^{-1/4}\exp(-x^{2}/2)~H_{n}(x),}En=n+12 ,{\displaystyle E_{n}=n+{\tfrac {1}{2}}~,}whereHn(x) are theHermite 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, thefundamental solution (propagator) ofH −i∂t, the time-dependent Schrödinger operator for this oscillator, simply boils down to theMehler kernel,[15][16]⟨x∣exp⁡(−itH)∣y⟩≡K(x,y;t)=12πisin⁡texp⁡(i2sin⁡t((x2+y2)cos⁡t−2xy)) ,{\displaystyle \langle x\mid \exp(-itH)\mid y\rangle \equiv K(x,y;t)={\frac {1}{\sqrt {2\pi i\sin t}}}\exp \left({\frac {i}{2\sin t}}\left((x^{2}+y^{2})\cos t-2xy\right)\right)~,}whereK(x,y;0) =δ(x −y). The most general solution for a given initial configurationψ(x,0) then is simplyψ(x,t)=∫dy K(x,y;t)ψ(y,0).{\displaystyle \psi (x,t)=\int dy~K(x,y;t)\psi (y,0)\,.}

Coherent states

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Main article:Coherent state
Coherent state dynamics forα=10{\displaystyle \alpha ={\sqrt {10}}}, in units of the harmonic oscillator lengthx0=ℏ/mω{\displaystyle x_{0}={\sqrt {\hbar /m\omega }}}, showing the probability density|ψ(x,t)|2{\displaystyle |\psi (x,t)|^{2}} and the quantum phase (color).

Thecoherent states (also known as Glauber states) of the harmonic oscillator are special nondispersivewave packets, with minimum uncertaintyσxσp =ℏ⁄2, whoseobservables'expectation values evolve like a classical system. They are eigenvectors of the annihilation operator,not the Hamiltonian, and form anovercomplete basis which consequentially lacks orthogonality.[17]

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

|α⟩=∑n=0∞|n⟩⟨n|α⟩=e−12|α|2∑n=0∞αnn!|n⟩=e−12|α|2eαa†e−α∗a|0⟩.{\displaystyle |\alpha \rangle =\sum _{n=0}^{\infty }|n\rangle \langle n|\alpha \rangle =e^{-{\frac {1}{2}}|\alpha |^{2}}\sum _{n=0}^{\infty }{\frac {\alpha ^{n}}{\sqrt {n!}}}|n\rangle =e^{-{\frac {1}{2}}|\alpha |^{2}}e^{\alpha a^{\dagger }}e^{-{\alpha ^{*}a}}|0\rangle .}

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 parameterα instead:α(t)=α(0)e−iωt=α0e−iωt{\displaystyle \alpha (t)=\alpha (0)e^{-i\omega t}=\alpha _{0}e^{-i\omega t}}.|α(t)⟩=∑n=0∞e−i(n+12)ωt|n⟩⟨n|α⟩=e−iωt2e−12|α|2∑n=0∞(αe−iωt)nn!|n⟩=e−iωt2|αe−iωt⟩{\displaystyle |\alpha (t)\rangle =\sum _{n=0}^{\infty }e^{-i\left(n+{\frac {1}{2}}\right)\omega t}|n\rangle \langle n|\alpha \rangle =e^{\frac {-i\omega t}{2}}e^{-{\frac {1}{2}}|\alpha |^{2}}\sum _{n=0}^{\infty }{\frac {(\alpha e^{-i\omega t})^{n}}{\sqrt {n!}}}|n\rangle =e^{-{\frac {i\omega t}{2}}}|\alpha e^{-i\omega t}\rangle }

Becausea|0⟩=0{\displaystyle a\left|0\right\rangle =0} and via the Kermack-McCrae identity, the last form is equivalent to aunitarydisplacement operator acting on the ground state:|α⟩=eαa^†−α∗a^|0⟩=D(α)|0⟩{\displaystyle |\alpha \rangle =e^{\alpha {\hat {a}}^{\dagger }-\alpha ^{*}{\hat {a}}}|0\rangle =D(\alpha )|0\rangle }. Calculating the expectation values:

⟨x^⟩α(t)=2ℏmω|α0|cos⁡(ωt−ϕ){\displaystyle \langle {\hat {x}}\rangle _{\alpha (t)}={\sqrt {\frac {2\hbar }{m\omega }}}|\alpha _{0}|\cos {(\omega t-\phi )}}

⟨p^⟩α(t)=−2mℏω|α0|sin⁡(ωt−ϕ){\displaystyle \langle {\hat {p}}\rangle _{\alpha (t)}=-{\sqrt {2m\hbar \omega }}|\alpha _{0}|\sin {(\omega t-\phi )}}

whereϕ{\displaystyle \phi } is the phase contributed by complexα. These equations confirm the oscillating behavior of the particle.

The uncertainties calculated using the numeric method are:

σx(t)=ℏ2mω{\displaystyle \sigma _{x}(t)={\sqrt {\frac {\hbar }{2m\omega }}}}

σp(t)=mℏω2{\displaystyle \sigma _{p}(t)={\sqrt {\frac {m\hbar \omega }{2}}}}

which givesσx(t)σp(t)=ℏ2{\textstyle \sigma _{x}(t)\sigma _{p}(t)={\frac {\hbar }{2}}}. Since the only wavefunction that can have lowest position–momentum uncertainty,ℏ2{\textstyle {\frac {\hbar }{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:ψα(x′)=(mωπℏ)14eiℏ⟨p^⟩α(x′−⟨x^⟩α2)−mω2ℏ(x′−⟨x^⟩α)2.{\displaystyle \psi _{\alpha }(x')=\left({\frac {m\omega }{\pi \hbar }}\right)^{\frac {1}{4}}e^{{\frac {i}{\hbar }}\langle {\hat {p}}\rangle _{\alpha }(x'-{\frac {\langle {\hat {x}}\rangle _{\alpha }}{2}})-{\frac {m\omega }{2\hbar }}(x'-\langle {\hat {x}}\rangle _{\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:

P(En)=|⟨n|α⟩|2=e−|α|2|α|2nn!{\displaystyle P(E_{n})=|\langle n|\alpha \rangle |^{2}={\frac {e^{-|\alpha |^{2}}|\alpha |^{2n}}{n!}}}

which corresponds to aPoisson distribution.

Highly excited states

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Wavefunction (top) and probability density (bottom) for then = 30 excited state of the quantum harmonic oscillator. Vertical dashed lines indicate the classical turning points, while the dotted line represents the classical probability density.

Whenn is large, the eigenstates are localized into the classical allowed region, that is, the region in which a classical particle with energyEn 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 throughasymptotics of the Hermite polynomials, and also through theWKB approximation.

The frequency of oscillation atx is proportional to the momentump(x) of a classical particle of energyEn and positionx. Furthermore, the square of the amplitude (determining the probability density) isinversely proportional top(x), reflecting the length of time the classical particle spends nearx. The system behavior in a small neighborhood of the turning point does not have a simple classical explanation, but can be modeled using anAiry function. Using properties of the Airy function, one may estimate the probability of finding the particle outside the classically allowed region, to be approximately2n1/332/3Γ2(13)=1n1/3⋅7.46408092658...{\displaystyle {\frac {2}{n^{1/3}3^{2/3}\Gamma ^{2}({\tfrac {1}{3}})}}={\frac {1}{n^{1/3}\cdot 7.46408092658...}}}This is also given, asymptotically, by the integral12π∫0∞e(2n+1)(x−12sinh⁡(2x))dx .{\displaystyle {\frac {1}{2\pi }}\int _{0}^{\infty }e^{(2n+1)\left(x-{\tfrac {1}{2}}\sinh(2x)\right)}dx~.}

Phase space solutions

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

The Wigner quasiprobability distribution for the energy eigenstate|n⟩ is, in the natural units described above,[18]Fn(x,p)=(−1)nπℏLn(2(x2+p2))e−(x2+p2),{\displaystyle F_{n}(x,p)={\frac {(-1)^{n}}{\pi \hbar }}L_{n}\left(2(x^{2}+p^{2})\right)e^{-(x^{2}+p^{2})}\,,}whereLn are theLaguerre polynomials. This example illustrates how the Hermite and Laguerre polynomials arelinked through theWigner map.

Meanwhile, theHusimi Q function of the harmonic oscillator eigenstates have an even simpler form. If we work in the natural units described above, we haveQn(x,p)=(x2+p2)nn!e−(x2+p2)π{\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 theSegal–Bargmann transform. Specifically, since theraising operator in the Segal–Bargmann representation is simply multiplication byz=x+ip{\displaystyle z=x+ip} and the ground state is the constant function 1, the normalized harmonic oscillator states in this representation are simplyzn/n!{\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

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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.[19]

N-dimensional isotropic harmonic oscillator

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The one-dimensional harmonic oscillator is readily generalizable toN dimensions, whereN = 1, 2, 3, .... In one dimension, the position of the particle was specified by a singlecoordinate,x. InN dimensions, this is replaced byN position coordinates, which we labelx1, ...,xN. Corresponding to each position coordinate is a momentum; we label thesep1, ...,pN. Thecanonical commutation relations between these operators are[xi,pj]=iℏδi,j[xi,xj]=0[pi,pj]=0{\displaystyle {\begin{aligned}{[}x_{i},p_{j}{]}&=i\hbar \delta _{i,j}\\{[}x_{i},x_{j}{]}&=0\\{[}p_{i},p_{j}{]}&=0\end{aligned}}}

The Hamiltonian for this system isH=∑i=1N(pi22m+12mω2xi2).{\displaystyle H=\sum _{i=1}^{N}\left({p_{i}^{2} \over 2m}+{1 \over 2}m\omega ^{2}x_{i}^{2}\right).}

As the form of this Hamiltonian makes clear, theN-dimensional harmonic oscillator is exactly analogous toN independent one-dimensional harmonic oscillators with the same mass and spring constant. In this case, the quantitiesx1, ...,xN would refer to the positions of each of theN particles. This is a convenient property of ther2 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}≡{n1,n2,…,nN}{\displaystyle \{n\}\equiv \{n_{1},n_{2},\dots ,n_{N}\}} the energy eigenfunctions for theN-dimensional oscillator are expressed in terms of the 1-dimensional eigenfunctions as:

⟨x|ψ{n}⟩=∏i=1N⟨xi∣ψni⟩{\displaystyle \langle \mathbf {x} |\psi _{\{n\}}\rangle =\prod _{i=1}^{N}\langle x_{i}\mid \psi _{n_{i}}\rangle }

In the ladder operator method, we defineN sets of ladder operators,

ai=mω2ℏ(xi+imωpi),ai†=mω2ℏ(xi−imωpi).{\displaystyle {\begin{aligned}a_{i}&={\sqrt {m\omega \over 2\hbar }}\left(x_{i}+{i \over m\omega }p_{i}\right),\\a_{i}^{\dagger }&={\sqrt {m\omega \over 2\hbar }}\left(x_{i}-{i \over m\omega }p_{i}\right).\end{aligned}}}

By an analogous procedure to the one-dimensional case, we can then show that each of theai andai† operators lower and raise the energy byℏω respectively. The Hamiltonian isH=ℏω∑i=1N(ai†ai+12).{\displaystyle H=\hbar \omega \,\sum _{i=1}^{N}\left(a_{i}^{\dagger }\,a_{i}+{\frac {1}{2}}\right).}This Hamiltonian is invariant under the dynamic symmetry groupU(N) (the unitary group inN dimensions), defined byUai†U†=∑j=1Naj†Ujifor allU∈U(N),{\displaystyle U\,a_{i}^{\dagger }\,U^{\dagger }=\sum _{j=1}^{N}a_{j}^{\dagger }\,U_{ji}\quad {\text{for all}}\quad U\in U(N),}whereUji{\displaystyle U_{ji}} is an element in the defining matrix representation ofU(N).

The energy levels of the system areE=ℏω[(n1+⋯+nN)+N2].{\displaystyle E=\hbar \omega \left[(n_{1}+\cdots +n_{N})+{N \over 2}\right].}ni=0,1,2,…(the energy level in dimension i).{\displaystyle n_{i}=0,1,2,\dots \quad ({\text{the energy level in dimension }}i).}

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

The degeneracy can be calculated relatively easily. As an example, consider the 3-dimensional case: Definen =n1 +n2 +n3. All states with the samen will have the same energy. For a givenn, we choose a particularn1. Thenn2 +n3 =n −n1. There aren −n1 + 1 possible pairs{n2,n3}.n2 can take on the values0 ton −n1, and for eachn2 the value ofn3 is fixed. The degree of degeneracy therefore is:gn=∑n1=0nn−n1+1=(n+1)(n+2)2{\displaystyle g_{n}=\sum _{n_{1}=0}^{n}n-n_{1}+1={\frac {(n+1)(n+2)}{2}}}Formula for generalN andn [gn being the dimension of the symmetric irreduciblen-th power representation of the unitary groupU(N)]:gn=(N+n−1n)=(N+n−1)!n!(N−1)!,n=n1+n2+n3+⋯.{\displaystyle g_{n}={\binom {N+n-1}{n}}={\frac {(N+n-1)!}{n!(N-1)!}},\;\;\;n=n_{1}+n_{2}+n_{3}+\cdots .}The special caseN = 3, given above, follows directly from this general equation. This is however, only true for distinguishable particles, i.e. in Maxwell-Boltzmann statistics (not in quantum statistics) or one particle inN dimensions (as dimensions are distinguishable). For the case ofN bosons in a one-dimension harmonic trap, the degeneracy scales as the number of ways to partition an integern using integers less than or equal toN. It can be shown that the large-E{\displaystyle E} asymptotic behavior of the degeneracygn{\displaystyle g_{n}} is practically independent of the energyE{\displaystyle E} - different from the classical case in which this diverges[20]. This degeneracy is

gn=p(N−,n).{\displaystyle g_{n}=p(N_{-},n).}

This arises due to the constraint of puttingN quanta into a state ket where∑k=0∞knk=n{\textstyle \sum _{k=0}^{\infty }kn_{k}=n} and∑k=0∞nk=N{\textstyle \sum _{k=0}^{\infty }n_{k}=N}, which are the same constraints as in integer partition.

Example: 3D isotropic harmonic oscillator

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See also:Particle in a spherically symmetric potential § 3D isotropic harmonic oscillator
Schrödinger 3D spherical harmonic orbital solutions in 2D density plots; theMathematica source code that used for generating the plots is at the top

The Schrödinger equation for a particle in a spherically-symmetric three-dimensional harmonic oscillator can be solved explicitly by separation of variables. This procedure is analogous to the separation performed in thehydrogen-like atom problem, but with a differentspherically symmetric potentialV(r)=12μω2r2,{\displaystyle V(r)={1 \over 2}\mu \omega ^{2}r^{2},}whereμ is the mass of the particle. Becausem will be used below for the magnetic quantum number, mass is indicated byμ, instead ofm, as earlier in this article.

The solution to the equation is:[21]ψklm(r,θ,ϕ)=Nklrle−νr2Lk(l+12)(2νr2)Ylm(θ,ϕ){\displaystyle \psi _{klm}(r,\theta ,\phi )=N_{kl}r^{l}e^{-\nu r^{2}}L_{k}^{\left(l+{1 \over 2}\right)}(2\nu r^{2})Y_{lm}(\theta ,\phi )}where

Nkl=2ν3π2k+2l+3k!νl(2k+2l+1)!!  {\displaystyle N_{kl}={\sqrt {{\sqrt {\frac {2\nu ^{3}}{\pi }}}{\frac {2^{k+2l+3}\;k!\;\nu ^{l}}{(2k+2l+1)!!}}}}~~} is a normalization constant;ν≡μω2ℏ {\displaystyle \nu \equiv {\mu \omega \over 2\hbar }~};
Lk(l+12)(2νr2){\displaystyle {L_{k}}^{(l+{1 \over 2})}(2\nu r^{2})}

aregeneralized Laguerre polynomials; The orderk of the polynomial is a non-negative integer, coinciding with the number of nodes of the radial part of the wavefunction;

The energy eigenvalue isE=ℏω(2k+l+32).{\displaystyle E=\hbar \omega \left(2k+l+{\frac {3}{2}}\right).}The energy is usually described by the singlequantum numbern≡2k+l.{\displaystyle n\equiv 2k+l\,.}

Becausek is a non-negative integer, for every evenn we haveℓ = 0, 2, ...,n − 2,n and for every oddn we haveℓ = 1, 3, ...,n − 2,n . The magnetic quantum numberm is an integer satisfying−ℓ ≤m ≤ℓ, so for everyn andℓ there are 2ℓ + 1 differentquantum states, labeled bym . Thus, the degeneracy at leveln is∑l=…,n−2,n(2l+1)=(n+1)(n+2)2,{\displaystyle \sum _{l=\ldots ,n-2,n}(2l+1)={(n+1)(n+2) \over 2}\,,}where the sum starts from 0 or 1, according to whethern is even or odd.This result is in accordance with the dimension formula above, and amounts to the dimensionality of a symmetric representation ofSU(3),[22] the relevant degeneracy group.

Applications

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Harmonic oscillators lattice: phonons

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See also:Canonical quantization

The notation of a harmonic oscillator can be extended to a one-dimensional lattice of many particles. Consider a one-dimensional quantum mechanicalharmonic chain ofN identical atoms. This is the simplest quantum mechanical model of a lattice, and we will see howphonons arise from it. The formalism that we will develop for this model is readily generalizable to two and three dimensions.As in the previous section, we denote the positions of the masses byx1,x2, ..., as measured from their equilibrium positions (i.e.xi = 0 if the particlei is at its equilibrium position). In two or more dimensions, thexi are vector quantities. TheHamiltonian for this system is

H=∑i=1Npi22m+12mω2∑{ij}(nn)(xi−xj)2,{\displaystyle \mathbf {H} =\sum _{i=1}^{N}{p_{i}^{2} \over 2m}+{1 \over 2}m\omega ^{2}\sum _{\{ij\}(nn)}(x_{i}-x_{j})^{2}\,,}wherem is the (assumed uniform) mass of each atom, andxi andpi are the position andmomentum operators for thei th atom and the sum is made over the nearest neighbors (nn). However, it is customary to rewrite the Hamiltonian in terms of thenormal modes of thewavevector rather than in terms of the particle coordinates so that one can work in the more convenientFourier space.

Superposition of three oscillating dipoles- illustrate the time propagation of the common wave function for different n,l,m

We introduce, then, a set ofN "normal coordinates"Qk, defined as thediscrete Fourier transforms of thexs, andN "conjugate momenta"Π defined as the Fourier transforms of theps,Qk=1N∑leikalxl{\displaystyle Q_{k}={1 \over {\sqrt {N}}}\sum _{l}e^{ikal}x_{l}}Πk=1N∑le−ikalpl.{\displaystyle \Pi _{k}={1 \over {\sqrt {N}}}\sum _{l}e^{-ikal}p_{l}\,.}

The quantitykn will turn out to be thewave number of the phonon, i.e. 2π divided by thewavelength. It takes on quantized values, because the number of atoms is finite.

This preserves the desired commutation relations in either real space or wave vector space

Another illustration of the time propagation of the common wave function for three different atoms emphasizes the effect of the angular momentum on the distribution behavior

[xl,pm]=iℏδl,m[Qk,Πk′]=1N∑l,meikale−ik′am[xl,pm]=iℏN∑meiam(k−k′)=iℏδk,k′[Qk,Qk′]=[Πk,Πk′]=0 .{\displaystyle {\begin{aligned}\left[x_{l},p_{m}\right]&=i\hbar \delta _{l,m}\\\left[Q_{k},\Pi _{k'}\right]&={1 \over N}\sum _{l,m}e^{ikal}e^{-ik'am}[x_{l},p_{m}]\\&={i\hbar \over N}\sum _{m}e^{iam(k-k')}=i\hbar \delta _{k,k'}\\\left[Q_{k},Q_{k'}\right]&=\left[\Pi _{k},\Pi _{k'}\right]=0~.\end{aligned}}}

From the general result∑lxlxl+m=1N∑kk′QkQk′∑leial(k+k′)eiamk′=∑kQkQ−keiamk∑lpl2=∑kΠkΠ−k ,{\displaystyle {\begin{aligned}\sum _{l}x_{l}x_{l+m}&={1 \over N}\sum _{kk'}Q_{k}Q_{k'}\sum _{l}e^{ial\left(k+k'\right)}e^{iamk'}=\sum _{k}Q_{k}Q_{-k}e^{iamk}\\\sum _{l}{p_{l}}^{2}&=\sum _{k}\Pi _{k}\Pi _{-k}~,\end{aligned}}}it is easy to show, through elementary trigonometry, that the potential energy term is12mω2∑j(xj−xj+1)2=12mω2∑kQkQ−k(2−eika−e−ika)=12m∑kωk2QkQ−k ,{\displaystyle {1 \over 2}m\omega ^{2}\sum _{j}(x_{j}-x_{j+1})^{2}={1 \over 2}m\omega ^{2}\sum _{k}Q_{k}Q_{-k}(2-e^{ika}-e^{-ika})={1 \over 2}m\sum _{k}{\omega _{k}}^{2}Q_{k}Q_{-k}~,}whereωk=2ω2(1−cos⁡(ka)) .{\displaystyle \omega _{k}={\sqrt {2\omega ^{2}(1-\cos(ka))}}~.}

The Hamiltonian may be written in wave vector space asH=12m∑k(ΠkΠ−k+m2ωk2QkQ−k) .{\displaystyle \mathbf {H} ={1 \over {2m}}\sum _{k}\left({\Pi _{k}\Pi _{-k}}+m^{2}\omega _{k}^{2}Q_{k}Q_{-k}\right)~.}

Note that the couplings between the position variables have been transformed away; if theQs and Πs wereHermitian (which they are not), the transformed Hamiltonian would describeNuncoupled harmonic oscillators.

The form of the quantization depends on the choice of boundary conditions; for simplicity, we imposeperiodic boundary conditions, defining the(N + 1)-th atom as equivalent to the first atom. Physically, this corresponds to joining the chain at its ends. The resulting quantization is

k=kn=2nπNafor n=0,±1,±2,…,±N2.{\displaystyle k=k_{n}={2n\pi \over Na}\quad {\hbox{for}}\ n=0,\pm 1,\pm 2,\ldots ,\pm {N \over 2}.}

The upper bound ton comes from the minimum wavelength, which is twice the lattice spacinga, as discussed above.

The harmonic oscillator eigenvalues or energy levels for the modeωk areEn=(12+n)ℏωkforn=0,1,2,3,…{\displaystyle E_{n}=\left({1 \over 2}+n\right)\hbar \omega _{k}\quad {\hbox{for}}\quad n=0,1,2,3,\ldots }

If we ignore thezero-point energy then the levels are evenly spaced at0, ℏω, 2ℏω, 3ℏω, ⋯{\displaystyle 0,\ \hbar \omega ,\ 2\hbar \omega ,\ 3\hbar \omega ,\ \cdots }

So anexact amount ofenergyħω, must be supplied to the harmonic oscillator lattice to push it to the next energy level. In analogy to thephoton case when theelectromagnetic field is quantised, the quantum of vibrational energy is called aphonon.

All quantum systems show wave-like and particle-like properties. The particle-like properties of the phonon are best understood using the methods ofsecond quantization and operator techniques described elsewhere.[23]

In thecontinuum limit,a → 0,N → ∞, whileNa is held fixed. The canonical coordinatesQk devolve to the decoupled momentum modes of a scalar field,ϕk{\displaystyle \phi _{k}}, whilst the location indexi (not the displacement dynamical variable) becomes the parameterx argument of the scalar field,ϕ(x,t){\displaystyle \phi (x,t)}.

Molecular vibrations

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Main article:Molecular vibration

Hooke's law

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  • TheHooke's atom is a simple model of thehelium atom using the quantum harmonic oscillator.
  • Hooke's law models models a mass moving on a spring where the force acting on the mass is proportional to its displacement.[11]
  • The general solution for a mass on a spring can be derived from this assumption.[11]
  • The displacement of the mass reaches a maxima and a minima at A and -A where A is called the amplitude.
  • This system is considered a conservative system where total energy remains unchanged and is being continuously redistributed between kinetic and potential energy.

The inverted harmonic oscillator

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The inverted harmonic oscillator has been investigated in detail by G. Barton.[25] See also H.J.W. Müller-Kirsten[26] and C. Yuce, A. Killen and A. Coruh.[27]

The Dirac oscillator

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The consideration of the harmonic oscillator e.g. from the energyE=p2/2m+mω2q2/2{\displaystyle E={\mathbf {p}}^{2}/2m+m\omega ^{2}q^{2}/2} in analogy to a derivation of the Dirac equation – so-to-speak from the "square root" of the equationpμpμ+m2=0{\displaystyle p_{\mu }p^{\mu }+m^{2}=0} – has been explored by Lorella M. Jones.[28]

See also

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Notes

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  1. ^The normalization constant isC=(mωπℏ)1/4{\displaystyle C=\left({\frac {m\omega }{\pi \hbar }}\right)^{{1}/{4}}}, and satisfies the normalization condition∫−∞∞ψ0(x)∗ψ0(x)dx=1{\displaystyle \int _{-\infty }^{\infty }\psi _{0}(x)^{*}\psi _{0}(x)dx=1}.

References

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  1. ^Griffiths 2004.
  2. ^Liboff 2002.
  3. ^Rashid, Muneer A. (2006)."Transition amplitude for time-dependent linear harmonic oscillator with Linear time-dependent terms added to the Hamiltonian"(PDF).M.A. Rashid –Center for Advanced Mathematics and Physics.National Center for Physics. Archived fromthe original(PDF-Microsoft PowerPoint) on 3 March 2016. Retrieved19 October 2010.
  4. ^Harald J.W. Müller-Kirsten, Introduction to Quantum Mechanics: Schrödinger Equation and Path Integral, 2nd ed. (2012) World Scientific, ISBN 978-9810-4397-5. This reference includes in detail the method of operators, Hermite functions, contour integration, pp. 130-141.
  5. ^Zwiebach (2022), pp. 233–234.
  6. ^Zwiebach (2022), p. 234.
  7. ^Zwiebach (2022), p. 241.
  8. ^Gbur, Gregory J. (2011).Mathematical Methods for Optical Physics and Engineering. Cambridge University Press. pp. 631–633.ISBN 978-0-521-51610-5.
  9. ^Zwiebach (2022), p. 240.
  10. ^Zwiebach (2022), pp. 249–250.
  11. ^abcdeMcQuarrie, Donald (7 August 2023).Physical Chemistry: A Molecular Approach (1st ed.). University Science Books. pp. 157–173.ISBN 9781891389962.
  12. ^Zwiebach (2022), pp. 246–249.
  13. ^Zwiebach (2022), p. 248.
  14. ^Hall, Brian C. (2013),Quantum Theory for Mathematicians, Graduate Texts in Mathematics, vol. 267, Springer, Theorem 11.4,Bibcode:2013qtm..book.....H,ISBN 978-1461471158
  15. ^Pauli, W. (2000),Wave Mechanics: Volume 5 of Pauli Lectures on Physics (Dover Books on Physics).ISBN 978-0486414621 ; Section 44.
  16. ^Condon, E. U. (1937). "Immersion of the Fourier transform in a continuous group of functional transformations",Proc. Natl. Acad. Sci. USA23, 158–164.online
  17. ^Zwiebach (2022), pp. 481–492.
  18. ^Cerisola, Federico; Mayo, Franco; Roncaglia, Augusto J. (11 October 2023)."A Wigner Quasiprobability Distribution of Work".Entropy.25 (10): 1439.arXiv:2303.08755.Bibcode:2023Entrp..25.1439C.doi:10.3390/e25101439.ISSN 1099-4300.PMC 10606729.PMID 37895560.
  19. ^Harald J.W. Müller-Kirsten, Introduction to Quantum Mechanics: Schrödinger Equation and Path Integral, 2nd ed. (2012), World Scientific, pp. 111-114.
  20. ^Harald J.W. Müller-Kirsten, Basics of Statistical Physics, 3rd ed.(2022), World Scientific, ISBN 978-9810-125-609-7, pp. 203-204.
  21. ^Albert Messiah,Quantum Mechanics, 1967, North-Holland, Ch XII,  § 15, p 456.online
  22. ^Fradkin, D. M. (1965). "Three-dimensional isotropic harmonic oscillator and SU3".American Journal of Physics.33 (3):207–211.doi:10.1119/1.1971373.
  23. ^Mahan, GD (1981).Many particle physics. New York: Springer.ISBN 978-0306463389.
  24. ^"Quantum Harmonic Oscillator".Hyperphysics. Retrieved24 September 2009.
  25. ^G. Barton. The inverted harmonic oscillator, Ann. Phys. (N.Y.) 166 (1986) 322.
  26. ^Harald J. Müller-Kirsten, Introduction to Quantum Mechanics: Schrödinger Equation and Path Integral, 2nd ed. (2012) World Scientific, pp. 130-141.
  27. ^C. Yuce, A. Killen and A. Coruh, Inverted Oscillator, quant-ph/0703234.
  28. ^Lorella M. Jones, Another Dirac Oscillator, University of Illinois at Urbana-Champaign report ILLL-(TH)-91-24 (1991).

Bibliography

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External links

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