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Kenneth M. SenadosMS Physics IPhys241 Quantum Mechanics 1

Quantum Behavior

Atomic Mechanics

Things on the atomic scale behave nothing like the ordinary. In fact, things on thisscale as Feynman describes is an unlikely ordinary experience which appears mysteriousto everyone both to the novice and experienced physicist.We all know very well how bigthings like balls and cars act, but things on the atomic scale dont act on the same way.Things on this scale is impossible to describe in the classical way. They do not behavelike particles or waves or like anything.

In 1926 and 1927, Schrodinger,Heisenberg, and Bohr resolved the confusion on atomicbehaviors by obtaining a consistent description of matter on the atomic scale. QuantumMechanics describes the behavior of matter and light in all details, in particular to thehappenings on atomic scale. The obtained description applies to all atomic objects.What apples to electrons applies to protons, neutrons, photons and the like.

An experiment with bullets and waves

The electrons were thought at first as particles and then later it was found outthat in many aspects it behaves like waves so it really behaves like neither. To try to

understand the quantum behavior of electrons, an experimental setup with the moreparticular behavior of particles like bullets, and with the behavior of waves like waterwaves is presented here.

Figure 1: Interference experiment with bullets.

Consider and ideal experimental setup shown in Figure 1 which describes the behav-ior of bullets and answers the question on what the probability is that a bullet whichpasses through the holes will arrive at the backstop and be detected by the detector.

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The setup contains a machine gun which fires indestructible bullets randomly over afairly large angular speed. In front it is a wall that has two holes just big enough for thebullets to pass through. Beyond the wall is a backstop which absorbs the bullet when

they hit it. It also has a detector of bullets that can be moved back and forth in the x-axis.

We shall say that these bullets alwaysarrive in indentical lumps. What is measurewith the detector is the probability of arrival of a lump and is measured as a function ofx. The result is plotted in the graph drawn in part (c) of Figure 1. We call the probabil-ityP12 because the bullets may have come through either hole 1 or 2. P12 is large nearthe middle of the graph but gets small as x gets large. However we may wonder whyP12 has its maximum at x = 0. This can be understood by covering one of the holes ata time. The maximum ofP1 occurs at the value ofx which is on the straight line withthe gun and hole 1. This is also true for the maximum of the probability P2. Comparingparts (b) and (c) of Figure 1, we see a very important result that

P12= P1+P2. (1)

The effect of both holes open is the sum the effect with each hole open alone. Thebullets come in lumps and their probability shows no interference.

Now consider again a experimental set-up shown in Figure 2 which describes thebehavior of waves.The setup is composed of a shallow trough of water which will be thewater source jiggled up and down by a motor and makes circular waves.To the right ofthe detector is a wall with two holes and beyond is a second wall which is the absorberwith a detector attached that can be moved back and forth in the x-axis.The detectormeasures the wave intensity.

Figure 2: Interference experiment with waves.

With this wave apparatus, it is observed that the intensity of the wave vary. Theintensity can have any value at all. There was no lumpiness in the wave inten-

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sity.Measuring the wave intensity for various values of x, we get the curve marked I12 inthe figure. In this case we would observed that the original wave is diffracted at the holesand new circular waves spread out the hole. However, if we cover one hole at a time and

measure the wave intensity in the absorber, we see a rather simple curve shown in part(b) of the figure. The intensity I12 is definitely not the sum ofI1 and I2. We see thatthere are series of constructive and destructive interference of the two waves. The waves

interfere destructivelyat places where the two waves arrive in the detector with phasedifference of . The low values ofI12 refer to where the waves interfere destructively.

The quantitative relationship between I1, I2, and I12 and the proper relation forinterferingwaves are

I1=|h1|2, I2=|h2|

2, I12 =|h1+h2 |2 . (2)

which is very different from the results obtained with the bullets. Expanding| h1+h262we see that

|h1+h2|2=|h1

2+| h2+ 2 | h1| h2cos (3)

where is the phase difference between h1 and h2. In terms of the intensities, we couldwrite

I12 = I1+I2+ 2

I1I2cos. (4)

where the last term is the interference term.

The experiment with electrons

Now the same experiment discussed previously is done with the electrons. The setupis described below:

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Figure 3: Interference experiment with electrons.

Noticed from the electron experiment is a sound of clicks from the detector andall these clicks are the same. However, these clicks come very erratically the same

as what is heard in a geiger counter. As the detector is moved around, it is noticed thatthe intensity of the sound varies but the loudness stays the same. It is also noticed thatif there are two detectors,one or the other will click, but not both at once. Thus, we cansay that electrons arrive in lumps.

Now we can answer the question on the probability of the arrival of the electron atthe backstop. The result of the experiment is an interesting curved markedP12in Figure3.

From the observations in the experiments, we can say that each electron either goesthrough hole 1 or it goes through hole 2. However,the result of the probability P12

is clearly not the same with the result that we have with the experiment of bullets.Instead, P12 is comparable to the one we got with the water waves. Thus we say thereis interference. For electrons,

P12=P1+P2. (5)

which is quite mysterious. The relation ofP1 and P2 with P12 can be described by twocomplex numbers that we can call 1 and 2. The mathematics for P12 is the same forthe water waves. Thus we say that

P12=|1+2 |2 . (6)

We conclude that the electrons arrive in lumps like particles and the probability ofthe arrival of these lumps is distributed like the distribution of the intensity of a wave.Now, since it is not true that P12=P1+ P2 then electrons neither go through hole 1 orhole 2.

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Another experiment was done in order to see in which hole the electrons pass through.We may add a light source behind the walls in the middle of the two holes and observewhere the electrons pass through. In this setup however, it is surprising that we get a

different result. We see that the probability that an electron will arrive at the backstopit similar to the experiment with bullets. We no longer get the old interference curveP12 but if we turn off the light, we get again P12.If the electrons are not seen, we haveinterference. The light (whatever its intensity or frequency is)might have disturb thebehavior of the electrons. In this experiment, it was found out that it is impossible toobserve as to which hole the electron went through without disturbing its pattern.

Heisenberg proposed his uncertainty principle which can state the experiments de-scribe as follows: It is impossible to design an apparatus to determine which hole theelectron went through without disturbing the electrons enough to destroy the interference

pattern.

This is quantum mechanics. Quite different with what we know in classical mechan-ics. In the experiment with electrons, it is impossible to predict exactly what wouldhappen. What has been talk about here are only probabilities that a certain event willhappen.

The uncertainty principle

Heisenbergs uncertainty principle tells us that if we make the measurement of anyobject,and we can determine the x-component its momentum with an uncertainty p,

we cannot, at the same time, know its x-position more accurately than x = h/pwherehis the Plancks constant. The uncertainties of the momentum and position musthave their product greater than Plancks constant. The uncertainty principle protectsquantum mechanics. If it was possible that we measure the momentum and positionwith greater accuracy, then quanum mechanics would collapse. With these, quantummechanics was able to live but with much mystery.

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