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=                    Quantum harmonic oscillator                     =
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                             Introduction
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The quantum harmonic oscillator is the quantum-mechanical analog of
the classical harmonic oscillator.   Because an arbitrary potential
can usually 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.


 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 position 'x'. Note: The
graphs are not normalized, and the signs of some of the functions
differ from those given in the text.
Corresponding probability densities.

The Hamiltonian of the particle is:
:\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  is the particle's mass,  is the force constant, \omega =
\sqrt{\frac{k}{m}} is the angular frequency of the oscillator, \hat{x}
is the position operator (given by ), 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 its potential energy, as in
Hooke's law.

One may write the time-independent Schrödinger equation,
:: \hat H \left| \psi \right\rangle = E \left| \psi \right\rangle  ~,
where  denotes a to-be-determined real number that will specify a
time-independent energy level, or eigenvalue, and the solution
denotes that level's energy eigenstate.

One may solve the differential equation representing this eigenvalue
problem in the coordinate basis, for the wave function , using a
spectral method. It turns out that there is a family of solutions. In
this basis, they amount to  Hermite functions,
:  \psi_n(x) = \frac{1}{\sqrt{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 'Hn' are the physicists' Hermite polynomials,
:H_n(z)=(-1)^n~ e^{z^2}\frac{d^n}{dz^n}\left(e^{-z^2}\right).

The corresponding energy levels are
: E_n = \hbar \omega \left(n + {1\over 2}\right)=(2 n + 1) {\hbar
\over 2} \omega~.

This energy spectrum is noteworthy for three reasons.  First, the
energies are quantized, meaning that only discrete energy values
(integer-plus-half multiples of ) 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  state, called the ground state)
is not equal to the minimum of the potential well, but  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 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
========================
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 position  and brighter colors represent
higher probability densities.

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  and its adjoint ,
:\begin{align}
a &=\sqrt{m\omega \over 2\hbar} \left(\hat x + {i \over
m \omega} \hat p \right) \\
a^\dagger &=\sqrt{m\omega \over 2\hbar} \left(\hat x - {i \over
m \omega} \hat p \right)
\end{align}

This leads to the useful representation of \hat{x} and  \hat{p},
:\begin{align}
\hat x &=  \sqrt{\frac{\hbar}{2}\frac{1}{m\omega}}(a^\dagger +
a) \\
\hat p &= i\sqrt{\frac{\hbar}{2}m\omega}(a^\dagger - a) ~.
\end{align}

The operator  is not Hermitian, since itself and its adjoint  are not
equal. The energy eigenstates , when operated on by these ladder
operators, give
:\begin{align}
a^\dagger|n\rangle &= \sqrt{n + 1} | n + 1\rangle \\
a|n\rangle &= \sqrt{n} | n - 1\rangle.
\end{align}

It is then evident that , in essence, appends a single quantum of
energy to the oscillator, while  removes a quantum. For this reason,
they are sometimes referred to as "creation" and "annihilation"
operators.

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

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
:\hat H = \hbar\omega\left(N + \frac{1}{2}\right),

so the eigenstate of  is also the eigenstate of energy.

The commutation property yields
:\begin{align}
Na^{\dagger}|n\rangle &= \left(a^\dagger N + [N,
a^\dagger]\right)|n\rangle \\
&= \left(a^\dagger N +
a^\dagger\right)|n\rangle \\
&= (n + 1)a^\dagger|n\rangle,
\end{align}

and similarly,
:Na|n\rangle = (n - 1)a | n \rangle.

This means that  acts on    to produce, up to a multiplicative
constant,   , and  acts on    to produce . For this reason,  is called
a annihilation operator ("lowering operator"), and  a creation
operator ("raising operator"). The two operators together are called
ladder operators. In quantum field theory,  and  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, , to produce another eigenstate with  less energy. By
repeated application of the lowering operator, it seems that we can
produce energy eigenstates down to . However, since
: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 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
:\hat H \left|0\right\rangle = \frac{\hbar\omega}{2}
\left|0\right\rangle