Glauber States: Coherent states of Quantum Harmonic Oscillator (2010 version)
Schrödinger's coherent states were discovered by Schrödinger in 1926. However, since then
the Born probability distribution has not yet been established, they have been largely
forgotten
Today, these states of harmonic oscillator, so called coherent states, are known as
Glauber states (R.J. Glauber: Nobel Prize, October 2005),
and they are known as single harmonic oscillator prototypes of
the coherent states of oscillating electromagnetic field, which are
the model of ideal continuous wave laser field, or an electromagnetic field
of unmodulated radio wave.
The probability distribution
of the coherent state behaves as the n=0 state whose shape moves as a classical
oscillator with the frequency omega.
In the toy below about 25 first states of harmonic oscillator are used
when in the coherent state mode, i.e. from the start on, or when you use the coherent state slider Energy. If you select or adjust any amplitudes in the left part, only the states 0 to 9 are involved and the amplitudes displayed (no phasing). The small numbered squares give you the single unmixed states.
To explore the behaviour of the Glauber State Toy, try the following:
- The Run/Stop button puts on animation. In some cases the animation stands still. That is
a correct behaviour (see single states). When running, it shows stop. When stopped, it shows Run,
and extra time arrows for single steps.
-
Select single states - pushbuttons upper left (Indicator text: Single State)
-
Select a combination of states - whenever the short vertical sliders change the amplitudes
(see below, the theory). The choice of a combination switches off the Glauber state precalculation.
(Indicator text: Superposed State)
-
The energy slider close to the display field: Choose the energy of the Glauber State.
Use of the Energy slider changes to Glauber states. (Indicator text: Glauber State)
Instructions:
-
The oscillator starts with amplitudes corresponding to a coherent state
with the shown energy (average energy).
-
The amplitudes are normalized. Thus, when only one amplitude is
nonzero, the wave function square (the probability density) remains unchanged.
- The wavefunction is shown at the position of the expectation value
of the energy
- Try to find the combination of amplitudes which forms the coherent Glauber state.
Hint: find in litterature the formula. They behave as
where
- Pressing "Start" and "Stop" changes the state of the animation
- When the animation is stopped, one can still use the arrow-buttons to
step through time.
The latest version of the code for the applett is available
here - Glauber2010.java
The original java applet has been written by Ondrej Psencik
(Quantum Harmonic Oscillator),
During the last 10 years
it has been much re-written and modified
to show better the properties of the
coherent states.
The latest major modifications are made to honnor The 2005 Nobel Prize winner R.J. Glauber
Quite important but smaller changes made July 2010 - 2010 visit of Nobel Prize winner R.J. Glauber in Bergen
The new development since spring 2000
is by L. Kocbach.
The functionality of the applet has also been implemented in 2002
in CERN ROOT environment - downlad and run in ROOT:
Glauber.C - or just read it as text: Glauber.C.txt