"Truth is stranger than fiction; fiction has to make sense."
- Leo Rosten
Quantum Mechanics: An Introduction
Part 1 of "An Introduction to Quantum Reality"
At the smallest, subatomic level, quantum mechanics has made remarkable discoveries about the behaviour of elementary particles. It would seem logical to assume that by studying these fundamental building blocks of our universe we might gain some of our best insights into the true nature of reality.
The predictions of quantum mechanics are very much driven by experimental results. Quantum mechanics does not have much to say about "why" things happen, it can just be used to predict "how" things behave, that behaviour being based on the well-established results of experiments. And the most famous of those experiments is the double-slit experiment.
In September 2002, the double-slit experiment was voted "the most beautiful experiment" by readers of Physics World (see here). Richard Feynman is said to have remarked that it contains everything you need to know about quantum mechanics. But perhaps the most useful property of the experiment is that it shows just how weird quantum reality can be!
In the double-slit experiment an electron gun is aimed at a screen with two slits, and the positions of the electrons is recorded after they pass through one of the two slits, making little dots on the screen. It is found that an interference pattern is produced on the screen just like the one produced by diffraction of a light or water wave passing through the two slits. There are bright bands ("constructive interference") and dark bands ("destructive interference").
This is quite strange: electron particles are interfering with each other as if they were waves. However, things turn much more weird if we only emit one electron at a time. We wait until an electron makes a dot on the screen before emitting another electron, so there is only ever one electron in the system at a time. A pattern slowly builds on the screen, dot-by-dot, as each individual electron hits the screen. What we see is quite incredible: the accumulation of dots on the screen eventually produces a pattern of light and dark bands - an interference pattern emerges even though there is only one electron in the system at any one time! The electron appears to be interfering with itself. So in some strange way it is as if the electron goes through both slits at once!!
How can this be? Maybe half the electron is going through one slit and half through the other slit? But if we put a small detector screen on the other side of one slit we only detect whole electrons sometimes passing through (sometimes passing through the other slit). It's as if the electron does indeed pass through both slits at once, but when we make an attempt to detect it, it suddenly decides to act like a single particle which has gone through just one slit!
It is only in the last ten years or so that quantum mechanics has at last been able to shed some light on what is happening in the double-slit experiment (this will be considered later in the page on Quantum Decoherence), but before we get to that stage there is a fair bit of theory to be covered. And it's best to start at the beginning.
Quantum mechanics could be said to have started in 1900 when Max Planck made the discovery that light, which was considered to be purely wave-like, was in fact composed of energy which came in discrete packets (called "quanta"). In the Planck formula, the energy of the packets, e, is proportional to the light frequency, f, the constant of proportionality being Planck's constant, h:
This result suggested that waves (light) were in fact composed of particles. The converse of this result
came in 1923 when Louis de Broglie (pronounced to rhyme with "destroy") suggested that matter (particles) behaves as a wave
(as is evident in the double-slit experiment), the wavelength, 
,
being inversely proportional to the particle's momentum, p. Here's the derivation:
We now know that absolutely everything in the known universe is made out of these strange particle/wave entities which obey these two formulae for quantum behaviour, given above.
How can we make sense of this strange wave/particle duality? Would it be possible to combine these two results in a single equation, and in the process reveal more insights into the true nature of reality at the quantum level?
Yes, Erwin Schrödinger did it in 1926. And we'll see that in the next page on The Quantum Casino.
The Heisenberg Uncertainty Principle
Now let us imagine that we want to measure the position a particular particle. In order to do this, we must "see" the particle, so
we have to shine some light of wavelength, , on it. This means there
is an inevitable uncertainty about the position of the particle (due to the resolving power of the light used), the uncertainty in the
particle's position being:
When the light particle (photon) hits the particle under observation it inevitable alters its momentum (speed) according to the result of Louis de Broglie (given above):
and on combining these two equations we get:
which is the Heisenberg Uncertainty Principle.
The Heisenberg Uncertainty Principle tells us that the more accurately we determine a particle's position,
, the less accurately we can know its momentum,
. This Uncertainty Principle equation
has been used to generate the animation below:
The animation shows the relevant spreads in the uncertainty for position and momentum (for a light wave, and the light wave's corresponding photon particle).
From the result of de Broglie (considered above) we know that if we have a precise value for a particle's momentum, p, then
we have a precise value for the wavelength of the wave, , and hence
a precise colour of our light wave. A precise value for particle momentum is shown in the above animation when the momentum value becomes a single vertical
point. You will see that when that occurs, the corresponding uncertainty in the particle's position spreads out to infinity, i.e., when we have light
behaving as a normal light wave (with a colour) we can have no idea where the corresponding particle is. In the wave/particle duality model this shows the
case when the light is acting like a wave, not a particle.
Conversely, the situation when we have a precise position for the particle, e.g., when we shine the light wave onto a screen and detect the proton, is shown in the above animation when the position value becomes a single vertical point. You will see that occurs, the corresponding uncertainty in the particle's momentum spreads out to infinity, i.e., we can now no longer determine the wavelength or the colour of the light wave. In the wave/particle duality model this shows the case when the light is acting like a particle, not a wave.
Comments
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So these elementary particles behave very differently to "ordinary objects". They behave according to quantum mechanical rules: sometimes they behave like a wave, sometimes like a particle. The more we constrain one property (say, the position property) to act like a particle, localised in space, the more the other property (momentum) acts like a wave, spread out in space.
Wikipedia provides a helpful analogy with sound waves which resembles your "two points in space" idea: "It is meaningless to ask about the frequency spectrum at a single moment in time, because the measure of frequency is the measure of a repetition recurring over a period of time. Indeed, in order for a signal to have a relatively well-defined frequency, it must persist for a long period of time, and conversely, a signal that occurs at a relatively well-defined moment in time (i.e., of short duration) will necessarily encompass a broad frequency band." See http://en.wikipedia.org/wiki/Uncertainty_principle So the more you try to pin-down the frequency of a wave, you need to stretch out your time measurements (the two position measurements you mention), and the more accurately you know the time, the less accurate is the frequency measurement.
But thanks for your idea. You'd be correct for everyday objects, just not for quantum mechanical objects. - Andrew Thomas, 11th September 2007
I'm just curious: there is no info about who you are and what is your work and why you do this kind of mathematical charity ?!
All comments date from few months ago. Hv u started this site so recently?
Hugs
Francisco Peres, 48, Curitiba - Brasil
francperes@uol.com.br - Francisco Peres, 4th November 2007
I am so glad I found this web site because you present the material so that I can understand it.
It really gets me thinking alright.
A real good starting point for my study of Quantum Mechanical Concepts.
I am fond of saying that we are all here because we probably can be because I believe this to be true.
Well--that is all I have to say.
Bye.
Tiffy - Tiffany, 19th March 2008
And here I am, taking notes!...
cheers,
1/f )))
http://www.algomantra.com/ - DJ Fadereu, 21st March 2008
Your derivation of the uncertainity principle is very counter-intuitive. I'm not being able to understand how you derived -
p = h/lambda
=> delta p = h/delta lamba
this seems incorrect (from a calculus viewpoint)
Wouldn't an approach using operators be a more correct approach (although, it will be a more mathematically rigorous approach)? - Kedar, 17th May 2008
The thing is, this website is not about being mathematically rigorous. Buy a textbook if you want that. I'm trying to explain quantum mechanics to everyone, and show that even someone with school-level maths can follow the derivation of the Uncertainty Principle! Hence showing that quantum mechanics is nothing to be scared of. Thanks for your comment. - Andrew Thomas, 17th May 2008
E=hf and p=h/lambda
were known to be true for photons. de Broglie then POSTULATED that these relations hold also for material particles. See Tipler for an explanation of how he came to make these postulates.
For a material particle, the relationship between the REST MASS of a particle and its energy is a little complicated. Even if you use the relativistic mass, m, when, indeed,
E=mc^2,
then still
p = m u,
where u is the velocity of the particle. In particular,
u<c
and
p<mc.
See Tipler for more details. - Dick, 19th May 2008
Cheers, Dick.
P.S. In response to Kedar's query:
Actually,
p = h/lambda
=> delta p = delta(h/lamba) = h*delta(1/lamba)
Now,
delta(1/lamba) = -(1/(lambda^2))*delta(lambda)
Well, approximately, as this is only strictly true if delta is an infinitesimal! - Dick, 21st May 2008
p = m u < m c
i.e., p = mc ONLY for massless particles such as the photon. p is not equal to mc for a material particle with non-zero rest mass.
Also for light of a definite wavelength, lambda, i.e., for a plane wave, Delta x is actually infinite, not lambda. And, in this case, Delta p is zero, as p has the definite value,
p = h/lambda
The rigorous way to derive the Uncertainty Relation is to do it for a wave packet. As I suggest, have a look at Tipler's "Physics" or similar simple text, if interested. Although in Tipler the general wave result for a wave packet is just quoted! But, it does give you an extra factor of 1/(2*pi). Mathematically, there's a further factor of 1/2 (see Wikipedia on the Uncertainty Principle). However, this might be a good case where it might have been best to just quote the result and used your wave pictures in momentum and in position to illustrate the Principle. Cheers:) - Dick, 22nd May 2008
Considering your first point, you're right about the energy of particles - I have modified the text to make it clear I am dealing with photons. Good spot, thanks. For the second point, we are taking a measurement of position of the particle (using incident light), so delta x is finite not infinite. So the lambda refers to the wavelength of the *incident* light used in the measurement, and it is the resolving power of that light which gives the delta x. That incident light then affects the momentum of the initial photon (so delta p is not zero). So it's the wavelength of the *incident* light which connects the change in position AND momentum of the initial photon.
I think it's a really simple and ingenious derivation, and I stand by it. People have nagged me before about my maths so hopefully I have ironed out all the remaining errors on the site. But thanks for the details in your own comment and the reference to Tipler's book - people can follow that up if they want rigour. On this site, though, I keep it simple and clear. - Andrew Thomas, 22nd May 2008
|psi(x)|^2,
where psi(x) is its wave function.
For a wave with the definite wavelength, lambda,
psi(x) = exp(i(kx-wt)),
where k is the wave number, 2*pi/lambda, and w is the angular frequency. In which case,
|psi(x)|^2 = 1,
whatever the x. Thus, there is an equal probability of finding the particle anywhere. Hence, the uncertainty in the particle position is infinite.
Just look at your own picture for a pure sinusoidal wave in x. In particular, the wave has a definite wavelength. There you can see that the particle position is very uncertain, but on the other hand the plot for p is just a spike, i.e., p is a definite number and so there is no uncertainty in p!
Cheers:) - Dick, 23rd May 2008
Consider a plane wave incident on an infinitely massive point object. Then, by observing the scattered spherical wave, the object's position is determined with certainty as being at the centre of the sphere of the scattered wave. Thus, in this case, Delta x = 0.
As I said, I don't see where in your thought experiment the instrinsic quantum mechanical nature of the object being observed comes in.
Cheers:) - Dick, 3rd June 2008
I agree about your second point, and that link I just posted says it well: "Looking closer at this picture, modern physicists warn that it only hides an imaginary classical mechanical interaction one step deeper, in the collision between the photon and the electron. In fact Heisenberg's microscope, although it was a big help in developing and teaching the quantum theory, is not itself part of current understanding. The true quantum interaction, and the true uncertainty associated with it, cannot be demonstrated with any kind of picture that looks like everyday colliding objects. To get the actual result you must work through the formal mathematics that calculates probabilities for abstract quantum states." - Andrew Thomas, 3rd June 2008
In fact, you could even propose your own interpretation of quantum mechanics, and - as long as it agreed with experimental observation - it would be just as valid as the existing interpretations.
I can recommend "Quantum" by Jim Al-Khalili as a good introductory book. - Andrew Thomas, 31st August 2008
I am astonished of how simple you explain quantum mechanics, I've never found something like that.
Now my problem is,
you say that delta(x) ~~ lambda
How do you get to this formula?
Thanks - Christoph, 10th October 2008
- Nils of Sweden, 20th October 2008
I haven't included a section on quantum computation as that is really an **application** of quantum mechanics, whereas this website deals with **fundamentals**. - Andrew Thomas, 20th October 2008
It is now very clear to me where Heisenberg and De Broglie erred.
The uncertainty principle only applies to measurement using electromagnetic waves. It implies that there is a limit to the resolving power of any measuring instrument based on electro-magnetism that can ever be build.
However, Heisenberg and De Broglie commit a logical fallacy in thinking that a particle cannot exist with a distinct position and momentum at the same time just because those parameter cannot be measured with accuracy at the same time using electromagnetism!
There could exist other measuring systems that do not rely on EM waves, maybe using dark energy or neutrinos.
I would sincerely appreciate it if you can comment on my thought (info (at) biotele (dot) com)
Thank you - biotele, 30th October 2008
However, there are still some concepts that I don't understand.
Where does the uncertainty principle apply to? Elementary particles and photons?
Why is it impossible to measure both position and momentum precisely? Is it because the observer changes the nature of things?
"when we have light behaving as a normal light wave (with a colour) we can have no idea where the corresponding particle is" -- I thought light was made of electric and magnetic fields only. So where does the particle you're talking about come in?
About the gifs you provided what are those waves refering to? Is it the space where you might find the particle within their envelope?
According to de Broglie postulate do particles whether behave literally like a wave or appear to act like one? It doesn't make sense to me imagine that a particle following a wave path will produce the same results as a real wave (ie. radiation)...
This kind of questions are poorly described in almost every resource (articles, books, etc), even from lecturers. That's why quantum mechanics is not a easy subject to grasp, at least for me as I like to understand things in detail. - Tiago, 2nd November 2008
You asked: '"Why is it impossible to measure both position and momentum precisely?" Well, if we knew the answer to that we would get a Nobel Prize. It's just the way the mathematics of quantum mechanics works. It's just the way the quantum world is.
You asked: "I thought light was made of electric and magnetic fields only. So where does the particle you're talking about come in?" Light can be considered as a particle OR a wave, the particle is a photon. You need to read about wave/particle duality: http://en.wikipedia.org/wiki/Wave-particle_duality
Particles behave like a REAL wave, not as a particle following a wave. It's all very strange! But that's quantum mechanics. - Andrew Thomas, 3rd November 2008
Imagining we were the same size as the smallest particles... Would we be able to measure them (position and momentum) precisely?
Do the particles always behave like a wave when they aren't under any measurements? Is that why the quantum world is probabilistic?
Are their wave-behaviour related to their wavefunction? - Tiago, 6th November 2008
You asked "Do the particles always behave like a wave when they aren't under any measurements?" Well, it's true to say that before we detect a particle's position it acts as though it is in a superposition state, so yes, it behaves like a wave. You then asked: "Is that why the quantum world is probabilistic?" Well, not really, but when we detect the particle's position we get a random result, so that is where the probability comes in. You then asked: "Are their wave-behaviour related to their wavefunction?" Well, yes, the wave behaviour IS related to the wavefunction in that two wavefunctions for two different particles can add together to create constructive or destructive interference (just like a wave). And after we know the shape of the final wavefunction, Max Born said the probability of finding the particle at a certain position was equal to the square of the wavefunction: http://tinyurl.com/6pn68v But the wavefunction is more than a simple probability wave as two wavefunctions can introduce interference effects (seen in the double-slit experiment) but probabilities can never be negative (and so can never produce interference). - Andrew Thomas, 6th November 2008
What do you mean with superposition state? You're saying that before any measurements a particle is spread in space so it doesn't have a well defined position only probabilities where it might be; therefore the particle seems to exist in different places at the same time? Is that when quantum mechanics gets bizarre? :P
I thought the wavefunction would give you the probability of finding a particle in x at a/any given time... I can't understand the difference between the wavefunction itself and its square... - Tiago, 6th November 2008
Moving on to consider the wavefunction, the wavefunction is a very peculiar entity which again has no real parallel in our everyday experience as we can only express it accurately by using mathematics - the phase of the wave is a complex number. This is described in more detail on the "Quantum Casino" page. But, yes, by squaring the wavefunction we convert the complex number to a real, normal number which we can take to be the probability of finding the particle. - Andrew Thomas, 7th November 2008
So let me see if now I can understand it. The uncertainty principle is a built-in uncertainty in nature so to speak. It also comes from the fact that the observer plays an essential role in the behaviour of particles. If we aren't taking measurements about the position of a particle, the particle itself is spread out over space (wave behaviour), and as we don't know its position the particle seems to exist in every possible position where it might occur (superposition). Therefore the better we know about its momentum (I'm following the maths but this seems to make no sense at all lol), because its wavelength is related to where the particle might be found. However, if we try to measure the position of a particle, the wavefunction (wave behaviour) collapses and the momentum/wavelength (again this only makes sense using the maths lol... momentum of the particle or momentum of where it might be found?) is no longer certain.
Did I mention something wrong? Am I in a good way about the understanding of quantum mechanics? - Tiago, 9th November 2008