Click here to go to the applet.
This java applet is a quantum mechanics simulation that shows the
interaction of classical electromagnetic radiation with
a particle, in one dimension. This applet will make more sense
if you have already used the Quantum States applet.
At the top of the applet you will see a graph of the potential, along
with horizontal lines showing the unperturbed energy levels. By default
it is an infinite square well (zero everywhere inside, infinite at the
edges). The electric field will cause the floor of the well to move
slightly; you will be able to see this movement better if you turn
up the radiation intensity.
Below the potential graph you will see an arrow showing the direction and
magnitude of the electric field, and the current caused by the
particle's motion. (The particle is assumed to be positively charged,
and the current is assumed to be proportional to
d<x>/dt.)
Below that you will see the probability distribution of
the particle's position. At the bottom of the screen is
a set of phasors showing the current particle state.
Underneath
the phasors is
set of circles showing possible final states; the blue color indicates
the transition rate to that state (assuming the correct
frequency is chosen). The darker the blue color,
the faster the transition; transitions which are extremely slow or
forbidden by selection rules are shown as black.
Between each graph is a horizontal line which may be dragged up and down
to adjust the size of each graph.
When the applet starts up, there is an incoming wave which is at
the right frequency (the Bohr frequency)
to transition the particle from the ground state to the first excited
state. Once it reaches the first excited state, it transitions back
to the ground state, and so on.
To view another transition, double-click the appropriate state
phasor to put the particle in that state. Then click the final
state underneath it. This will start an incoming wave with the right
frequency to make that transition. The initial and final states will
be shown in purple.
If you transition from a lower state to a higher state, that is called
absorption; the particle will absorb energy from the incoming wave.
In that case the current will be in phase with the electric field,
which would cause the wave to weaken somewhat in real life (but not in
this simulation).
If you transition from a higher state to a lower state, that is called
stimulated emission; the particle will lose energy. In that case the
current will be out of phase with the electric field,
which would cause the wave to get stronger (energy will be radiated away).
In order to simulate spontaneous emission, we need to quantize the
electromagnetic field. Since the wave in this applet is treated classically,
we can't do spontaneous emission.
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.
- Coupled Well Pair: this is two square wells with a
wall between them.
- Harmonic Oscillator: this is a harmonic oscillator
potential. This potential is unusual because the energy levels
are evenly spaced. So when transitioning from the ground state
to the first excited state, the particle will keep going into the
second excited state and then third excited state, etc., because the
Bohr frequency for the first two states is also the Bohr frequency
for the next two states. The effect of this is that the particle will
stay in a coherent state where it moves from side to side
without changing shape, and will
will keep acquiring more and
more energy, just like a classical particle in a forced harmonic oscillator
at resonance. (After a while the particle will go too far to the sides
and will become deformed; that is an artifact of this applet and would
not happen in a real harmonic oscillator.)
The Clear button clears out all states.
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 Stop Radiation button stops the incoming wave. The wave
will start again when you select a final state or adjust the
frequency slider.
The Reverse Phase button reverses the phase of the incoming
wave, which also reverses the direction of energy flow.
The Stopped checkbox stops the simulation temporarily.
The Simulation Speed slider changes the speed of the wave
function evolution.
The Radiation Intensity slider changes the intensity
of the incoming radiation. When the radiation is more intense,
transitions will proceed more quickly, but they might be
"messier" because of secondary transitions that would otherwise
be too weak to have an effect.
The Radiation Frequency slider changes the frequency
of the incoming radiation. Normally this is set automatically
by choosing the initial and final state, but it can also be set
manually. If you modify this at all, then you will not get exact
resonance and transitions will be much less likely to happen.
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 View Menu has the following items:
- Position: show the position graph (on by default)
- Momentum: show the momentum graph
- 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.
- 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.
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
A far higher place must be assigned to Judaism among the competitors for the allegiance of Europe. The cosmopolitan importance at one time assumed by this religion has been considerably obscured, owing to the subsequent devolution of its part to Christianity. It is, however, by no means impossible that, but for the diversion created by the Gospel, and the disastrous consequences of their revolt against Rome, the Jews might have won the world to a purified form of their own monotheism. A few significant circumstances are recorded showing how much influence they had acquired, even in Rome, before the first preaching of Christianity. The first of these is to be found in Cicero’s defence of Flaccus. 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. I was waiting there by my ship a good while back, and I heard another one cruising and spiraling, shooting the field, I guess, because he came in and set down. My crate, just the way you ordered, was down by the grove, not in plain sight in the middle of the course. But Jeff set his ship down, left the engine running, and went off. I stayed hid to see what would happen, but when he didn’t come back, I thought I’d better go and find you—and see if it meant anything to you.” She stood alone, with the sticky, wet knife in her hand, catching her breath, coming out of the madness. Then she stooped, and pushing the branches aside felt about for her pistol. It lay at the root of a tree, and[Pg 80] when she had picked it up and put it back in the holster, there occurred to her for the first time the thought that the shot in the dead stillness must have roused the camp. And now she was sincerely frightened. If she were found here, it would be more than disagreeable for Landor. 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. In presenting the money bills in the month of May, Speaker Onslow could not avoid remarking that there were two circumstances which tended to create alarm—foreign subsidies and foreign troops introduced, and nothing but their confidence in his Majesty could allay their fears, or give them confidence that their burdens would be soon reduced. There was, in fact, no chance for any such reduction, for wars, troubles, and disgraces were gathering around from various quarters. The first reverse came from the Mediterranean. MUCH to their amazement, the boys waked up the next morning in Nashville, and found that they had passed through the "dark and bloody ground" of Kentucky absolutely without adventure. After drawing and dividing the rations and cartridges. Si gave the boys the necessary instruction about having their things ready so that they could get them in the dark the next morning, and ordered them to disregard the bonfires and mirth-making, and lie down to get all the sleep they could, in preparation for the hard work of the next day. Then, like the rest of the experienced men, who saw that the campaign was at length really on, and this would be the last opportunity for an indefinite while to write, he sat down to write short letters to his mother and to Annabel. "Bully for the Wild Wanderers of the Wabash," Shorty joined in. "They're the boss regiment in the army o' the Cumberland, and the Army o' the Cumberland's the boss army on earth. Hooray for US Co. Le's have a speech. Where's Monty Scruggs?" "Bring a light, do—I can't abide this dark." Albert suddenly began to look uneasy. After all he was not really drunk, only a little fuddled. He walked straight, and his roll was natural to him, while though he was exceedingly cheerful, and often burst into song, his words were not jumbled, and he generally seemed to have a fair idea of what he was saying. "But I heard what the doctor said to you." "A purty accident—wud them stacks no more dry than a ditch. 'Twas a clear case of 'bustion—fireman said so to me; as wicked and tedious a bit o' wark as ever I met in my life." Calverley stept from the shadow of the cliff, and beheld a meteor in the sky, brightening and expanding, as the clouds opened, until it assumed the appearance of a brilliant star, of astonishing magnitude, encircled by dazzling rays, which, in a singular manner, were all inclined in one direction, and pointing to that part of the horizon where lay the rival of England—France. The foreman's face assumed a deeper hue than usual: he looked fiercely at the galleyman, but there was a determination in the weather-beaten face that made him pause ere he spoke. "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.