Why is probability amplitude squared?

1. Nov 24, 2013

Jilang

Does anyone have an intuitive understanding of why probabilities in QM are the amplitudes multiplied by their compex conjugates?

2. Nov 24, 2013

Staff: Mentor

I thought it was to give you a real value that could be measurable. The wiki article may explain it better:

http://en.wikipedia.org/wiki/Born_rule

or maybe not but at least there's some references to follow up.

Last edited: Nov 24, 2013
3. Nov 24, 2013

Jilang

I can see it gets rid of the phase, but the unitary evolution of the wavefunction applies to the amplitude not the probability so it must be more than just an mathematical convenience I think.

4. Nov 24, 2013

Jilang

Thanks Jedi, I didn't get too far with Wiki as the maths got a bit tricky. But I just found this on this forum from 2009
http://en.wikiversity.org/wiki/Making_sense_of_quantum_mechanics/Principles_of_Quantum_Mechanics
Which I found very helpful. It suggests it is a joint probability between the particle and detector, a joint probability, which to a fair approximation would be the amplitude of the moving particle multiplied by the complex conjugate. This makes some sort of sense, but I don't know if it's generally accepted so would appreciate any comments on this.

5. Nov 24, 2013

Bill_K

Feynman says this is the fundamental difference between classical and quantum mechanics. Quantum mechanics predicts probability amplitudes which are complex numbers, and the addition of complex numbers in which their relative phase plays a role is the root cause of all interference phenomena.

6. Nov 24, 2013

Jilang

Thank Bill, I am happy with that, but that is not what I am asking about.

7. Nov 24, 2013

audioloop

and more interesting, superposition is limited on space tieme

8. Nov 24, 2013

Jilang

Er ?? . Please expand.

9. Nov 24, 2013

Mandragonia

There is also a historical reason why QM became developed in this direction.

Early in the twentieth century it became apparent that electrons behave more like waves than like particles. Their behaviour is not well described by classical mechanics. Thus there was a need among physicists to develop a theory that accounted for the waviness of particles such as the electron. Now nobody really had a clue how to do this... So what they did, is borrow ideas and techniques from the -by far- most successful wave-theory at that time: Classical Electromagnetism. This theory had developed over many years, and culminated in the 4 Maxwell equations.

The complex wave function in the Schroedinger equation was adapted from the (complex) electric and magnetic fields of EM. The probability density is analogous to the intensity of the electromagnetic field (the absolute value of the electric field squared plus the absolute value of the magnetic field squared). In doing so, the QM theory incorporated the wave-effects of diffraction and interference that were observed for electrons and light, and which had been successfully described for the latter by the Maxwell equations.

10. Nov 24, 2013

Hypersphere

While one can go as deeply into the formalism as one likes, the best non-technical explanation I've heard is that the Schrödinger equation is a wave equation. For all waves, the amplitude squared gives an intensity. In quantum mechanics the "intensity" is the probability of finding the particle in a particular position, i.e. Schrödinger's equation describes some kind of probability wave for the particle.

11. Nov 24, 2013

audioloop

that probabilities vanishes with distance.
simply describes a observed fact but does not explain it.

.

12. Nov 24, 2013

Staff: Mentor

Well its the easiest way of getting a positive number from a complex number, which you must have for probabilities.

The deep reason however lies in Gleason's Theorem which proves the only probability measure you can define on complex vector spaces is the usual Born rule from which the squaring follows as a special case.

In this connection you will probably find the following interesting which attempts to explain it at an intuitive level, and I think is the type of thing you are after:
http://www.scottaaronson.com/democritus/lec9.html

Your next question is probably - why complex numbers? Well that's the quantum mystery isn't it? Feynman actually sorted it out - if you don't have complex numbers then you do not get phase cancellation on all the possible paths a particle can take leaving only those of stationary action. In fact you can actually derive Schrodingers equation from the Hamiliton-Jacobi equation of Classical Mechanics if you do one simple thing - go to complex numbers:
http://arxiv.org/abs/1204.0653

But that 'simple' thing has very very deep consequences containing the rock bottom essence of QM. There are other reasons as well such as the very important Wigners Theorem requires complex numbers, but IMHO that you can derive Schrodinger's equation this way is quite startling.

These ideas have sort of been fermenting around for a while, then Lucien Hardy put it all together in a seminal paper developing QM from scratch:
http://arxiv.org/pdf/quant-ph/0101012.pdf

We see there are basically two reasonable ways of modelling physical systems using probability models - standard probability theory and QM.

That's likely the deepest reason of all.

Thanks
Bill

Last edited: Nov 24, 2013
13. Nov 24, 2013

Devils

Bill.

Are there macroscopic or other examples of QM that can be seen intuitively? Since the study of probability has been around for hundreds of years, were the properties of QM discovered in the past century or earlier? ie why wasn't QM math worked out in the 1800s.

14. Nov 24, 2013

Staff: Mentor

Group theory has been around for yonks as well. Yet the very elegant derivation of SR using groups wasn't discovered - the mathematical machinery was there - but the insight necessary to do it took many years of theoretical and experimental investigation.

Check out:
http://www.pnas.org/content/93/25/14256.full

It took the penetrating genius of Einstein to start us on the path to this modern view of symmetry in physics. But that was just the start - much work had to occur before it was sorted out. The mathematical machinery was there - but key insights weren't.

QM is weird - no denying it. Without experimental necessity no one in their right mind would have proposed it. But with that necessity on us, gradually, oh so gradually, exactly what is going on is clearer. That exploration is far from over, and expect further insights and developments, but progress has been made, so now we understand QM is basically one of two most reasonable probability models.

Even though probability has been around for yonks, the study of generalized probability models as a discipline in its own right is only recent. For this modern view to emerge that was required as well.

Thanks
Bill

Last edited: Nov 24, 2013
15. Nov 24, 2013

Staff: Mentor

I think that the 'intuitiveness' of QM can only be seen via mathematics.

Thanks
Bill

16. Nov 25, 2013

Jilang

Thanks for the links, they were most interesting. I thought the part about the mirror reflection was wonderful! I think I can now appreciate that the probability needs to derived this way to avoid those damnable discontinuous classical jumps! If I am reading it right whatever dimension N you are working in you require at least N+1 to avoid the jumps and make the transitions continuous and N+1 is the most economical. The recurrent theme appears to be assuming continuity of transition as an axiom, but I am still unsure as to why the continuity is important. Is it to do with reversibility?

17. Nov 25, 2013

Staff: Mentor

Ok - you need to understand some ideas from generalized probability models.

They concern themselves with systems that can be described, in a very general sort of way, by something called the state. You can read the link on Lucien Hardy's paper for the detail.

Of special interest are states that are called pure - they have a special mathematical property of being the 'fundamental' states in that all the other states are convex sums of them.

Throw a dice and you have 6 outcomes - these are the pure states of that probability model. In general standard probability theory does not have a continuum of pure states - handling continuous variables requires a bit of care with weird stuff like the Dirac Delta function - you need to read the literature on probability models to understand this. There is only a finite or countably infinite number of them so you cant continuously change from one to the other. Intuitively if you want to model systems like you find in physics, such as for example a particles position, and you want want states to be able to continuously change to other states, you run into problems with standard probability theory.

You see this with what's known as a Wiener process which uses probability to model the position of a particle. Things do not quite work out - the path turns out to actually be continuous but is very strange - its everywhere non-differentiable - not that nice mathematically. However if you do something really nutty and allow a Wiener process to be complex - lo and behold you actually get QM - which is a very weird but strange fact.

The deep reason this works is it allows the model to escape the inability to continuously change from one pure state to the other. If you allow this - lo and behold you get QM.

What Lucien Hardy showed is some very general and reasonable considerations show only two possibilities for modelling physical systems exist - QM and normal probability theory - with the difference being the 'continuous' behavior of pure states. But, as the example of a Wiener process shows, things like position don't quite work out with normal probability theory, so you need QM.

Interestingly further work has been done that shows the other special characteristic QM has is entanglement. Either continuous transformations of pure states or entanglement is enough to single out QM as the only reasonable model:
http://arxiv.org/abs/0911.0695

To me this suggests the rock bottom basic thing about QM that makes it - well QM - is entanglement. And indeed in modern times entanglement is now what is thought to be the fundamental explanation for how the classical world emerges.

If you want to pursue that further check out:
http://theoreticalminimum.com/courses/quantum-entanglement/2006/fall

Thanks
Bill

18. Nov 25, 2013

Jilang

Thanks for describing this so nicely. I think you can derive the EM wave equation from the Maxwell equations. Are there any equations analogous to them in QM for deriving the Shroedinger equation?

19. Nov 25, 2013

Staff: Mentor

The correct way to derive Schrodinger's equation is via symmetry - you will find the detail in Chapter 3 - Ballentine - QM - A Modern Development:
https://www.amazon.com/Quantum-Mechanics-A-Modern-Development/dp/9810241054

Also get a hold of a book by the great Lev Landau, IMHO an absolute classic, that will quite likely change your view of physics, called Mechanics:
https://www.amazon.com/Mechanics-Third-Edition-Theoretical-Physics/dp/0750628960

I will repeat one review, just because it really is shockingly true:

'If physicists could weep, they would weep over this book. The book is devastingly brief whilst deriving, in its few pages, all the great results of classical mechanics. Results that in other books take take up many more pages. I first came across Landau's mechanics many years ago as a brash undergrad. My prof at the time had given me this book but warned me that it's the kind of book that ages like wine. I've read this book several times since and I have found that indeed, each time is more rewarding than the last.

The reason for the brevity is that, as pointed out by previous reviewers, Landau derives mechanics from symmetry. Historically, it was long after the main bulk of mechanics was developed that Emmy Noether proved that symmetries underly every important quantity in physics. So instead of starting from concrete mechanical case-studies and generalising to the formal machinery of the Hamilton equations, Landau starts out from the most generic symmetry and dervies the mechanics. The 2nd laws of mechanics, for example, is derived as a consequence of the uniqueness of trajectories in the Lagragian. For some, this may seem too "mathematical" but in reality, it is a sign of sophisitication in physics if one can identify the underlying symmetries in a mechanical system. Thus this book represents the height of theoretical sophistication in that symmetries are used to derive so many physical results.'

The foundation of QM is two axioms you will find in Ballentine - the dynamics is really a consequence of symmetry.

As Landau shows the foundation of Classical Mechanics is the Principle Of Least Action - and again the dynamics follows from symmetry.

But wait - those two axioms in Ballentine imply in the classical limit the Principle Of Least Action.

So at rock bottom, classically, or in QM, symmetry is the real reason behind the dynamics.

By now, hopefully, you will have glimpsed, but only glimpsed, one of the deepest and most profound insights physics has revealed - symmetry is what really governs the world. Its not talked about much in the pop-sci press - it should be - it's startling and shocking - but you only appreciate it by studying the real deal.

I invite you, and anyone else, to take that journey.

Thanks
Bill

Last edited: Nov 25, 2013
20. Nov 25, 2013

Jilang

Symmetry is my weak spot so I have ordered the Ballentine. I will wrap it up for myself for Xmas, 25 quid well spent and I've started my Christmas shopping also-not bad for a days work!

21. Nov 25, 2013

stevendaryl

Staff Emeritus
I do not understand that paper. Equation 4.2 says:

$S = -i \hbar ln(\psi)$

Equation 4.6 says:

$(\frac{\partial S}{\partial x})^2 = -\frac{\hbar^2}{\psi} \frac{\partial^2}{\partial x^2} \psi$

But that doesn't seem correct. Let's take the ground state of the harmonic oscillator, which has the form: $\psi = e^{-\lambda x^2 - i \omega t}$

For this $\psi$, we have:

$S = -i \hbar (-\lambda x^2 - i \omega t)$
$(\frac{\partial S}{\partial x})^2 = - 4 \hbar^2 \lambda^2 x^2$
$-\frac{\hbar^2}{\psi} \frac{\partial^2}{\partial x^2} \psi = -\hbar^2 (-2\lambda + 4\lambda^2 x^2)$

So it's not true that
$(\frac{\partial S}{\partial x})^2 = -\frac{\hbar^2}{\psi} \frac{\partial^2}{\partial x^2} \psi$

22. Nov 25, 2013

Mandragonia

You are welcome. And you are right. From the Maxwell equations (which are first order differential equations) one can derive the EM wave equation (which is a second order differential equation).

The Schroedinger equation is a first order differential equation in time. At that time, everything being new, it was perfectly logical to try this first. On the left hand side of the equation, where in EM there are the spatial operators (the rotation and divergence), Erwin Schroedinger placed the Hamilton operator (which everyone was already familiar with from classical mechanics). The somewhat odd mixture of EM field theory with a bit of classical mechanics turned out to be a fantastically accurate equation for understanding the behaviour of electrons in atoms!

You ask whether there is something similar to the EM wave equation hidden in the Schroedinger equation. No, not really. It turns out things are somewhat different in QM. That is because the left hand side (with the Hamiltonian) is second order in space. So their is a mismatch between space and time derivatives. In fact, the Schroedinger equation describes the time evolution of an electron wave as a kind of diffusion process. Electron waves have a tendency to spread out.

When the Schroedinger approach turned out to be an overwhelming success, physicists were interested whether they could also incorporate relativistic effects. So they set out to construct a relativistic version of the Schroedinger equation. This time they took the (second order) EM wave equation as a starting point; this is understandable because is has the desired property of Lorentz invariance. The QM version of the EM wave equation is called the Klein-Gordon equation.

23. Nov 25, 2013

San K

Sometimes it feels as if space-time, matter, energy, dark energy, dark matter etc....are all inter-convertible....

on a separate note:

1. what is "linear objects"? ....as in below:

"If we keep in mind that state vectors represent ordinary linear objects, there is nothing mysterious about quantum physics"

by linear objects does the writer mean one dimensional objects?

2. Why is spin (as in angular momentum) considered the marriage of quantum mechanics and relativity?

Last edited: Nov 25, 2013
24. Nov 25, 2013

Jilang

I think it means they obey the rules of linear algebra like:
U + V = V + U
Etc.
They can have as many dimensions as you want. Do you know matrices?

Last edited: Nov 25, 2013
25. Nov 25, 2013

Jilang

Yes, I can't get it to work either.