Click here to go to the applet.
This java applet is a quantum mechanics simulation that shows the
behavior of a particle in a two dimensional harmonic oscillator.
At the top of the screen, you will see a cross section of the potential,
with the energy levels indicated as gray lines. The red line is
the expectation value for energy.
In the center of the applet, you will see the probability distribution of
the particle's position. The color indicates the phase. At the bottom
of the screen is
a set of phasors showing the magnitude and phase of some of the possible
states.
You may 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. When you move the mouse over a state,
it is highlighted in yellow, and other states with the same energy
are also highlighted.
You may create a gaussian wave packet by clicking and dragging on
the position graph.
Between each graph is a horizontal line which may be dragged up and down
to adjust the size of each graph.
The Mouse Popup determines what happens when the mouse is
clicked. The choices are:
- Create Gaussian: this allows you to create a gaussian distribution
on either the position or momentum graph. Click on the graph to locate
the center of the gaussian. After clicking, drag the mouse to select
the size of the distribution.
- Gaussian w/ Momentum: this allows you to create a gaussian
distribution
on the position graph, with some initial momentum. Click on the graph
to locate
the center of the gaussian. The size of the gaussian will be the same
size as the last time you used Create Gaussian. After clicking,
drag the mouse to select the initial momentum. Note that if you
drag too far, the gaussian may not be formed properly.
- Rotate Function: this allows you to rotate the wave
function with the mouse. This changes the state of the particle to a
new state which is at a different orientation. States with nx+ny > 13
cannot be rotated; this may cause a high-energy particle's position
distribution to abruptly change shape when you attempt to rotate it.
Also, if the aspect ratio is not set to 1, then rotation won't work.
- Translate Function: this allows you to move the wave
function around with the mouse. If you take a stationary state and
translate it slightly, then it will oscillate about its original
position. If you move a state too close to the edge of the screen,
then it may not translate properly.
- Scale Function: this allows you to resize the wave
function around with the mouse. If you take a stationary state and
scale it slightly, then it will oscillate about its original
size. If you try to scale a state too large or too small,
then it may not work properly.
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 Stopped checkbox stops the evolution of the wave function.
The Alternate Rendering checkbox is used to speed up
rendering, but it actually slows things down on some machines.
(Internally, it uses the MemoryImageSource class instead of drawing
a bunch of rectangles.) It should be set to the proper setting by
default, but try selecting or unselecting it to see if it speeds
things up.
The Simulation Speed slider changes the speed of the wave
function evolution.
The Brightness slider controls the brightness, just like on a
TV set.
The Resolution slider changes the resolution of the applet. There
are only two possible resolutions.
The Aspect Ratio slider will adjust the strength of the
"spring constant" in the x direction as compared to the y direction.
The View Menu has the following items:
- Energy: show the energy/potential graph (on by default)
- Position: show the position graph (can't be turned off)
- Linear Momentum: show the linear momentum graph
- Angular Momentum: show the angular momentum graph. This is
only available if the aspect ratio is 1. States with nx+ny > 13 do not
appear on the angular momentum graph.
- Rectangular State: show the set of rectangular state
phasors (on by default)
- Angular State: show the set of angular state
phasors, allowing you to pick states by angular momentum. This is only
available if the aspect ratio is 1.
- 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.
- Wave Function: display the wave function in one of three
ways: as a probability (magnitude squared), as a probability with the
phase shown using colors, 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 set of states with the same energy
(using the state coefficients to determine
probability) and putting the particle in those states.
- Measure Angular Momentum: take a measurement of the angular
momentum,
picking a random set of states with the same m
(using the state coefficients to determine
probability) and putting the particle in those states. Only
works if the aspect ratio is 1.
The Options Menu has the following items:
On the right side of the phasor panel, you will see a phasor all by
itself. This phasor's rotation has the same period as the
oscillation of the wave function. Create a gaussian to see this. It
is only present if the aspect
ratio is square.
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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The latter was accused of appropriating part of the annual contributions sent to the temple at Jerusalem; and, in dealing with this charge, Cicero speaks of the Jews, who were naturally prejudiced against his client, as a powerful faction the hostility of which he is anxious not to provoke.330 Some twenty years later, a great advance has been made. Not only must the material interests of the Jews be respected, but a certain conformity to their religious prescriptions is considered a mark of good breeding, In one of his most amusing satires, Horace tells us how, being anxious to shake off a bore, he appeals for help to his friend Aristius Fuscus, and reminds him of217 some private business which they had to discuss together. 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The same poet elsewhere intimates that whenever they found themselves in a majority, they took advantage of their superior strength to make proselytes by force.’332 And they pursued the good work to such purpose that a couple of generations later we find Seneca bitterly complaining that the vanquished had given laws to the victors, and that the customs of this abominable race were established over the whole earth.333 Evidence to the same effect is given by Philo Judaeus and Josephus, who inform us that the Jewish laws and customs were admired, imitated, and obeyed over the whole earth.334 Such assertions might be suspected of exaggeration, were they not, to a certain extent, confirmed by the references already quoted, to which others of the same kind may be added from later writers showing that it was a common practice among the Romans to abstain from work on the Sabbath, and even to celebrate it by praying, fasting, and lighting lamps, to visit the synagogues, to study the law of Moses, and to pay the yearly contribution of two drachmas to the temple at Jerusalem.335 Jeff’s hand was quietly coming down. “What happened to you?” begged Sandy. “Something new has come up, sir. 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They must not find her. She started at a swift, long-limbed run, making a wide detour, to avoid the sentries, bending low, and flying silently among the bushes and across the shadowy sands. The year 1756 opened with menaces to England of the most serious nature. The imbecility of the Ministry was beginning to tell in the neglect of its colonies and its defences. France threatened to invade us, and a navy of fifty thousand men was suddenly voted, and an army of thirty-four thousand two hundred and sixty-three of native troops; but as these were not ready, it was agreed to bring over eight thousand Hessians and Hanoverians. To pay for all this it was necessary to grant excessive supplies, and lay on new duties and taxes. 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"Galleyman," he at length said, "you knew the business before you came: if you be so fond of saving old witches' lives, why didn't you say so, that I might not now be in this dilemma?" "No, no, not the boy," replied Merritt, rather impatiently.