## probability distribution

Matter and photons are waves, which means that they extend a certain distance. what is the position of a particle, such as an electron? is it in the center of the wave? the answer lies in how the position of an electron is measured. experiments show that it will find the electron somewhere definite, unlike a wave. but if you set up exactly the same situation and measure it again, you will find the electron in a different location, often far removed from any experimental uncertainty in your measurement. repeated measurements will show a statistical distribution of locations that appears as a wave. (see figure 1.)

After de Broglie proposed the wave nature of matter, many physicists, including Schrödinger and Heisenberg, explored the consequences. The idea quickly arose that, due to its wave-like nature, the trajectory and fate of a particle cannot be accurately predicted for each particle individually. however, each particle goes to a defined place (as illustrated in figure 1). after collecting enough data, it obtains a distribution related to the wavelength and diffraction pattern of the particle. there is a certain probability of finding the particle at a given location, and the overall pattern is called a probability distribution. those who developed quantum mechanics devised equations that predicted the probability distribution under various circumstances.

It is somewhat unsettling to think that you cannot predict exactly where an individual particle will go, or even follow it to its destination. Let’s explore what happens if we try to track a particle. Consider the double-slit patterns obtained for electrons and photons in Figure 2. First, we note that these patterns are identical, following d sin θ = mλ, the equation for double-slit constructive interference developed on photon energies and the electromagnetic spectrum, where d is the slit separation and λ is the wavelength of the electron or photon.

Both patterns statistically accumulate as individual particles fall onto the detector. this can be observed for photons or electrons; for now, let’s focus on electrons. you can imagine electrons interfering with each other like waves do. to test this, you can turn down the intensity until there is never more than one electron between the slits and the screen. the same interference pattern accumulates! this implies that the probability distribution of a particle spans both slits and the particles actually interfere with each other. does this also mean that the electron goes through both slits? an electron is a basic unit of matter that is not divisible. but it’s a fair question, so we should look to see if the electron goes through one slit or the other, or both. one possibility is to have coils around the slits that detect charges moving through them. what is observed is that an electron always passes through one slit or the other; it does not split to go through both. but there is a catch. if you determine that the electron went through one of the slits, you no longer get a double-slit pattern; instead you get a single slit interference. there is no escape using another method to determine which slit the electron passed through. knowing that the particle passed through a slit forces a single-slit pattern. if you don’t watch which slit the electron goes through, you’ll get a double-slit pattern.

## heisenberg uncertainty

How does knowing which slit the electron passed through change the pattern? The answer is fundamentally important: the measurement affects the system being observed. information can be lost, and in some cases it is impossible to measure two physical quantities simultaneously with exact precision. for example, you can measure the position of a moving electron by scattering light or other electrons from it. those probes themselves have momentum, and by scattering off the electron, they change their momentum in a way that loses information. there is a limit to absolute knowledge, even in principle.

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It was Werner Heisenberg who first established this limit of knowledge in 1929 as a result of his work on quantum mechanics and the wave characteristics of all particles. (see figure 3). specifically, consider simultaneously measuring the position and momentum of an electron (it could be any particle). there is an uncertainty in position Δx that is approximately equal to the wavelength of the particle. that is, Δx ≈ λ.

As discussed above, a wave is not located at a point in space. if the position of the electron is repeatedly measured, a scatter in the locations will be observed, implying an uncertainty in the position Δx. to detect the position of the particle, we must interact with it, such as making it collide with a detector. in the collision, the particle will lose momentum. this change in momentum could be anywhere from near zero to the total momentum of the particle, [latex]p=frac{h}{lambda}\[/latex]. it is not possible to know how much momentum will be transferred to a detector, so there is also an uncertainty in the momentum Δp. in fact, the uncertainty in momentum can be as large as the momentum itself, which in equation form means that [latex]delta{p}approxfrac{h}{lambda}\[/latex] .

Uncertainty in position can be reduced by using a shorter wavelength electron, since Δx ≈ λ. but shortening the wavelength increases the uncertainty in momentum, since [latex]p=frac{h}{lambda}\[/latex]. conversely, the uncertainty in momentum can be reduced by using a longer wavelength electron, but this increases the uncertainty in position. Mathematically, you can express this tradeoff by multiplying the uncertainties. the wavelength cancels, leaving ΔxΔp ≈ h.

then if one uncertainty decreases, the other must increase for their product to be ≈h.

using advanced mathematics, heisenberg showed that the best that can be done in a simultaneous measurement of position and momentum is [latex]delta{x}delta{p}gefrac{h}{4 pi}\[/latex].

this is known as the heisenberg uncertainty principle. it is impossible to measure position x and momentum p simultaneously with uncertainties Δx and Δp multiplying to be less than [latex]frac{h}{4pi}\[/latex]. nor can the uncertainty be zero. Neither uncertainty can be made small without the other being made large. a short wavelength allows accurate position measurement, but increases the sonde’s momentum to the point of further perturbing the momentum of the system being measured. For example, if an electron is scattered from an atom and has a wavelength small enough to detect the position of the electrons in the atom, its momentum can knock the electrons out of their orbits so that information about their location is lost. original movement. therefore, it is impossible to follow an electron in its orbit around an atom. if you measure the position of the electron, you will find it in a definite place, but the atom will break apart. repeated measurements on identical atoms will produce interesting probability distributions for the electrons around the atom, but will not produce motion information. the probability distributions are called electron clouds or orbitals. the shapes of these orbitals are often shown in general chemistry texts and discussed in the wave nature of quantization of matter causes.

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why don’t we notice the heisenberg uncertainty principle in everyday life? the answer is that Planck’s constant is very small. therefore, the lower bound on the uncertainty of measuring the position and momentum of large objects is negligible. we can detect sunlight reflected off jupiter and follow the planet in its orbit around the sun. reflected sunlight alters jupiter’s momentum and creates an uncertainty in its momentum, but this is totally insignificant compared to jupiter’s enormous momentum. the correspondence principle tells us that the predictions of quantum mechanics become indistinguishable from classical physics for large objects, as is the case here.

## heisenberg uncertainty for energy and time

Another form of the Heisenberg uncertainty principle exists for simultaneous measurements of energy and time. in equation form, [latex]delta{e}delta{t}gefrac{h}{4pi}\[/latex], where Δe is the uncertainty in energy and Δt is the uncertainty in time . this means that within a time interval Δt, it is not possible to measure the energy accurately; there will be an uncertainty Δe in the measurement. to measure the energy more accurately (to make Δe smaller), we must increase Δt. this time interval can be the amount of time it takes to perform the measurement, or it could be the amount of time that a particular state exists, as in example 2 below.

The uncertainty principle for energy and time can be of great importance if the lifetime of a system is very short. then Δt is very small and consequently Δe is very large. some nuclei and exotic particles have extremely short lifetimes (as little as 10−25 s), leading to energy uncertainties of up to many gev (109 ev). the stored energy appears as a larger rest mass, so this means that there is a significant uncertainty in the rest mass of short-lived particles. when repeatedly measured, a spread of masses or decay energies is obtained. the dispersion is Δe. You might wonder if this uncertainty in energy could be avoided by not measuring lifetime. the answer is no. nature knows the lifetime, so its brevity affects the particle’s energy. this is so well established experimentally that the uncertainty in the decay energy is used to calculate the lifetime of short-lived states. some nuclei and particles are so short-lived that it is difficult to measure their lifetime. but if its decay energy can be measured, its dispersion is Δe, and this is used in the uncertainty principle [latex]left(delta{e}delta{t}gefrac{h}{4 pi} right)\[/latex] to calculate the lifetime Δt.

There is another consequence of the uncertainty principle for energy and time. if the energy is uncertain by Δe, then the conservation of energy can be violated by Δe for a time Δt. neither the physicist nor nature can say that the conservation of energy has been violated, if the violation is temporary and less than the uncertainty in energy. while this sounds innocuous enough, we’ll see in later chapters that it allows for the temporary creation of matter out of thin air and has implications for how nature transmits forces over very small distances.

Finally, note that in the discussion of particles and waves, we have stated that individual measurements produce precise or particle-like results. a definite position is determined each time we observe an electron, for example. but repeated measurements produce a scatter in the values consistent with wave characteristics. the great theoretical physicist richard feynman (1918-1988) commented, “there are particles”. when you observe enough of them, they are distributed as you would expect for a wave phenomenon. however, what there is while they travel we cannot say because, when we try to measure, we affect the trip.

## section summary

- matter is found to have the same interference characteristics as any other wave.
- There is now a probability distribution for the location of a particle instead of a defined position.
- another consequence of the wave nature of all particles is the heisenberg uncertainty principle, which limits the precision with which certain physical quantities can be known simultaneously. for position and momentum, the uncertainty principle is [latex]delta{x}delta{p}gefrac{h}{4pi}\[/latex], where Δx is the uncertainty in the position and Δp is the uncertainty in momentum.
- for energy and time, the uncertainty principle is [latex]delta{e}delta{t}gefrac{h}{4pi}\[/latex] where Δe is the uncertainty in energy yΔt is the uncertainty in time.
- These small bounds are fundamentally important on the quantum mechanical scale.

## glossary

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**heisenberg uncertainty principle:** a fundamental limit to the precision with which pairs of quantities (momentum and position, and energy and time) can be measured

**energy uncertainty:** lack of precision or lack of knowledge of precise results in energy measurements

**time uncertainty:** lack of precision or lack of knowledge of precise results in time measurements

**impulse uncertainty:** lack of precision or lack of knowledge of precise results in impulse measurements

**position uncertainty:** lack of precision or lack of knowledge of precise results in position measurements

**probability distribution:** the general spatial distribution of probabilities of finding a particle at a given location