Click here to go to the applet.
This java applet is a quantum mechanics simulation that shows the
behavior of a single particle in bound states in one dimension. It solves
the Schrödinger equation and allows you to visualize the solutions.
At the top of the applet you will see a graph of the potential, along
with horizontal lines showing the energy levels. By default
it is an infinite well (zero everywhere inside, infinite at the
edges). Below that you will see the probability distribution of
the particle's position, oscillating back and
forth in a combination of two states. Below the particle's position
you will see a graph of its momentum. (This is just the fourier transform
of the particle's position.) At the bottom of the screen is
a set of phasors showing the magnitude and phase of all the possible
states.
To view a state, move
the mouse over its energy level on the potential graph. To select a
single state, click on it.
You may also select a single state by picking one of the phasors at the
bottom and double-clicking on it. Or, you may click on the phasor and
drag its value to modify the magnitude and phase. In this way, you can
create a combination of states. To delete a state, click on the phasor
and then drag the mouse far away from it.
Between each graph is a horizontal line which may be dragged up and down
to adjust the size of each graph.
Each graph also has a red line, indicating the expectation value
for that observable.
The Setup Popup allows you to select a predefined potential. The
choices are:
- Infinite Well: this is the "particle in a box"; the particle
is confined between two walls of infinite potential. The width of the
well is adjustable. Only a finite number of the states are shown;
increase the resolution to see more states.
- Finite Well: this is a square well of finite depth. Only
the bound states are shown in this applet. The width and depth of the
well are adjustable.
- Harmonic Oscillator: this is a harmonic oscillator
potential. The "spring constant" of the oscillator and its offset
are adjustable. (If you have a particle in a stationary state and then
translate it in momentum space, then the particle is
put in a coherent quasi-classical
state that oscillates like a classical particle.)
- Well Pair: this is two square wells of finite depth. The
width and separation of the two wells
are adjustable.
- Coupled Well Pair: this is two square wells with an adjustable
wall between them. The separation of the two wells and the
potential of the wall between them
are adjustable.
- Asymmetric Well: this is two attached square wells with
different widths and depths.
- Infinite Well + Field: this is an infinite square well
with a uniform electric field, which causes the potential to slope
downward. The width of the well and the field direction and strength
are adjustable.
- Coupled Wells + Field: this is two square wells with a
wall between them, in an electric field.
- Coulomb: this is vaguely similar to a coulomb potential,
except that it doesn't become infinite near the center, and it is
bounded on the left and right sides.
- Quartic Oscillator.
- Well Array (square): this is a series of square wells. The
number of wells and their depth are adjustable.
- Well Array (harmonic): this is a series of wells with a
harmonic oscillator shape.
- Well Array (coulomb): this is a series of wells with a
coulomb shape.
- Well Array + Field: this is a series of square wells with
an electric field.
- Well Array w/ Impurity: this is a series of square wells with
one well which has a different depth.
- Well Array w/ Dislocation: this is a series of square wells with
one well offset slightly.
- Random Well Array: this is a series of square wells with
random separations.
- 2 Well Array: this is a series of square wells with alternating
depths.
- Delta Fn Array: this is a set of up to 30 delta function
wells. (Actually they aren't true delta functions since the well width
and well depth are both finite, but the resulting wave functions look
like the theory predicts. The wells are as narrow as the applet's
resolution will allow.) The number of wells and their separation are
adjustable. If the number of wells and the well separation
are both too large, then not all the wells will fit on the screen.
The Mouse Popup determines what happens when the mouse is
clicked. The choices are:
- Set Eigenstate: if you click on the energy, position, or
momentum graph, then the particle will be put in an eigenstate of that
observable. So if you click on the energy graph, the particle will be
placed in a stationary state. If you click on the position graph,
the particle will be localized at that point (but will quickly spread
out). If you click on the momentum graph, the particle will have the
selected momentum.
It may not be possible to put the particle in the eigenstate you selected,
because this applet only deals with bound states, and there are a limited
number of them because of finite resolution. For example, you can't
put the particle outside a finite square well. If you try, then the
applet will just do the best it can.
Clicking on the position graph will cause the particle to be localized
as much as possible, which will often give unsatisfactory results because
the momentum spectrum will be so spread out. To
localize the particle with a little more uncertainty, use Edit Function
to sweep out an area of possible locations for the particle, or
use Create Gaussian to create a more spread-out distribution.
- Edit Function: this allows you to edit the potential energy
graph or position distribution by clicking and dragging on the appropriate
graph. When editing the position distribution, it's not possible to edit
the phase; the old phase is retained. (Again, it may not be possible to
make the changes you specified
to the particle's position distribution since it may require eigenstates
which are unbound.)
- Create Gaussian: this allows you to create a gaussian distribution
on either the position or momentum graph. The width of the distribution
can be controlled by moving the mouse up and down. (Again, it may not
be possible to make a well-formed gaussian because it may require
eigenstates that are unbound.)
- Translate Function: this allows you to drag the particle from
side to side, to change its position. You can also drag it in momentum space,
to change its momentum.
The Clear button clears out all states.
The Normalize button normalizes the set of particle
states. (By default, the states are not shown normalized because the
interface is easier to use if they are not. They are normalized internally
when calculating the wave functions, however.)
The Maximize button changes the magnitude of the
particle states so that they are all as large as possible. This makes
them easier to see. (It won't change the wave function at all
because the states are normalized internally.)
The Ground State button selects the ground state wave function.
The Rescale Graphs button changes the scale of all the graphs
so that everything is as large as possible. Normally, the scale is
adjusted only when necessary, so click this button if the wave functions
are too small to see clearly.
The Stopped checkbox stops the evolution of the wave function.
The Simulation Speed slider changes the speed of the wave
function evolution.
The Resolution slider changes the resolution of the applet.
The higher the resolution, the more accurate the wave functions and
energy levels will be.
The Particle Mass slider changes the mass of the particle.
The View Menu has the following items:
- Energy: show the energy/potential graph (on by default)
- Position: show the position graph (on by default)
- Momentum: show the momentum graph (on by default)
- Sum All States: show the sum of the probability distributions
of all states.
- Parity: show a graph of parity.
- Probability Current: show the probability current. This is
zero for stationary states. For states that are not stationary, the
probability current is positive where the wave function is moving
to the right and negative where it is moving to the left.
- Left/Right Waves: show the wave function decomposed into
two separate waves, one moving to the left (negative momentum) and
one moving to the right (positive momentum). This option only works with
the unmodified infinite well and harmonic oscillator potentials.
For best results,
select the Wave Function -> Magnitude and Phase menu item.
- Values/Dimensions: show some quantitative detail about the
potential and the wave function.
- State Phasors: show the set of state phasors (on by default)
- Expectation Values: show expectation values as red lines
- Uncertainties: show uncertainties as blue lines to the
left and right of the expectation value. The distance from the blue
lines to the red line is the uncertainty. This is not available on
the probability current graph, because that is not an observable.
- Wave Function: display the wave function in one of four
ways: as a probability (magnitude squared), as a probability with the
phase shown using colors, as real and imaginary parts graphed
separately, or as a magnitude with the phase shown using colors.
The Measure Menu has the following items:
- Measure Energy: take a measurement of the energy by
picking a random state (using the state coefficients to determine
probability) and putting the particle in that state.
- Measure Position: take a measurement of the position by
picking a random position (using the probability distribution)
and locating the particle at that point.
The Options Menu has the following items:
If you like this applet you may be interested in the book Visual Quantum Mechanics.
Click here to go to the applet.
java@falstad.com
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