Why is probability amplitude squared?

[Jilang]

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

 

[jedishrfu]

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

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.

 

[Jilang]

I thought it was to give you a real value that could be measurable.

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.

 

[Jilang]

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.

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.

 

[Bill_K]

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

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.

 

[Jilang]

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.

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

 

[audioloop]

and more interesting, superposition is limited on space tieme

 

[Jilang]

and more interesting, superposition is limited on space tieme

Er ?? . Please expand.

 

[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.

 

[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.

 

[audioloop]

Er ?? . Please expand.

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

 

[bhobba]

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

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

 

[Devils]

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

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.

 

[bhobba]

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.

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

 

[bhobba]

Are there macroscopic or other examples of QM that can be seen intuitively?

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

Thanks
Bill

 

[Jilang]

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

Thanks
Bill

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?

 

[bhobba]

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?

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 can't 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

 

[Jilang]

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.

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?

 

[bhobba]

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?

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/dp/9810241054/?tag=pfamazon01-20

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/dp/0750628960/?tag=pfamazon01-20

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

 

[Jilang]

The correct way to derive Schrodinger's equation...

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

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!

 

[stevendaryl]

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

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

 

[Mandragonia]

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?

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.

 

[San K]

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.

Good link.

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?

 

[Jilang]

Good link.

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

on a separate note:

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"

are linear objects (one or) two-dimensional objects?

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?

 

[Jilang]

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

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

 

[Jilang]

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

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.

I find this really fascinating. The Schroedinger Equation is a diffusion equation with an imaginary diffusion coefficient (or real diffusion in imaginary time?) Is that just a coincidence or is there some underlying process driving it? Why would it become more uncertain over time?

 

[bohm2]

I find this really fascinating. The Schroedinger Equation is a diffusion equation with an imaginary diffusion coefficient (or real diffusion in imaginary time?) Is that just a coincidence or is there some underlying process driving it? Why would it become more uncertain over time?

You might find this paper interesting:

From Classical Hamiltonian Flow to Quantum Theory: Derivation of the Schrödinger Equation
http://arxiv.org/ftp/quant-ph/papers/0311/0311109.pdf

 

[bhobba]

I do not understand that paper. Equation 4.2 says:

It has been a while since I have gone through that paper so I can't say off hand what's going on.

However in the derivation of the Schrodinger equation the action is transformed into a different form (the wavefunction) that can't be related to what it started with classically eg it may have solutions the original equation doesn't.

By this I mean you have shown the transformed action obeys a differential equation (Schrodinger's equation) so the transformed solution of the Hamilton-Jacobi should be a solution - if it isn't the derivation is incorrect - but it seems fine to me when I went through it. However the converse may not be true - ie a solution of Schrodinger's equation may not, when transformed back, solve the Hamilton-Jabobi.

Thanks
Bill

 

[bhobba]

I find this really fascinating. The Schroedinger Equation is a diffusion equation with an imaginary diffusion coefficient (or real diffusion in imaginary time?) Is that just a coincidence or is there some underlying process driving it? Why would it become more uncertain over time?

Feynman sorted it all out ages ago.

Its dead simple. If you go to complex numbers nearby paths cancel by differing in phase 180% - only those of stationary action remain.

The even deeper reason is not so simple - it has to do with having continuous transformations between pure states:
http://arxiv.org/pdf/quant-ph/0101012.pdf

Complex numbers allow you to do that - the pure states of ordinary probability theory don't.

Thanks
Bill

 

[bhobba]

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!

Its the best book on QM I know - by a LONG LONG way IMHO - and I have read a few.

Be warned however - its graduate level - meaning you need mathematical maturity.

If you find it a bit too hard going I would warm up with Lenny Susskinds books:
https://www.amazon.com/dp/046502811X/?tag=pfamazon01-20
https://www.amazon.com/dp/0465036678/?tag=pfamazon01-20

And the associated video lectures:
http://theoreticalminimum.com/

Thanks
Bill

 

[Mandragonia]

I find this really fascinating. The Schroedinger Equation is a diffusion equation with an imaginary diffusion coefficient (or real diffusion in imaginary time?) Is that just a coincidence or is there some underlying process driving it? Why would it become more uncertain over time?

1. Is it just coincidence?
Yes, I think so. The main focus of Erwin Schroedinger and his colleagues was on building a model that describes the atomic structure and its properties, in particular emission and absorption spectra. The behaviour of free electrons was (presumably) less important.

2. Is there some underlying process driving it?
The spreading of a wave packet is regarded a consequence of Heisenberg's uncertainty principle.

3. Why would it become more uncertain over time?
A free electron can be in sharply localized state at t=0, with small uncertainty in position dx and some uncertainty in velocity dv. If you wait for some time, the wave packet evolves. The uncertainty in position increases due to the additional uncertainty in the velocity. Finally it becomes of the order t*dv, which can be much larger than the initial uncertainty dx.

 

[Jilang]

1. Is it just coincidence?
Yes, I think so. The main focus of Erwin Schroedinger and his colleagues was on building a model that describes the atomic structure and its properties, in particular emission and absorption spectra. The behaviour of free electrons was (presumably) less important.

2. Is there some underlying process driving it?
The spreading of a wave packet is regarded a consequence of Heisenberg's uncertainty principle.

3. Why would it become more uncertain over time?
A free electron can be in sharply localized state at t=0, with small uncertainty in position dx and some uncertainty in velocity dv. If you wait for some time, the wave packet evolves. The uncertainty in position increases due to the additional uncertainty in the velocity. Finally it becomes of the order t*dv, which can be much larger than the initial uncertainty dx.

Thanks for this. Does the uncertainty in the velocity stay the same or does that increase too?

 

[Mandragonia]

Thanks for this. Does the uncertainty in the velocity stay the same or does that increase too?

It stays the same for a free electron. Of course in a different setting it may increase too. I forgot to mention that the technical term for the gradual spreading of the wave packet is dispersion.

 

[teodorakis]

Hi i want to replenish this thread.
The wave function (not squared) describes the probability of a particle occupying a particular location in space - but this needs to be multiplied by the probability of a particle being detected at that same location in space.

By way of analogy, if Alice and Bob can be at one of four places, then the probability of Alice (or Bob) being at anyone place is 1:4 (the wave function) and the probability of Alice meeting Bob is also 1:4, BUT the probability of Alice meeting Bob at a specificied location is 1:16, ie. 4 squared.

That's how I interpret the wave function squared.
Carl Looper
8 December 2009
This is Carl's way of explaining and since I'm a bit bad with advanced math. This seems the most intuitive explanation to me. Is it that simple or do ı have to learn all the hamiltonian spaces states etc. to fully undersstand why this amplitude is squared.
By the way, squared amplitude is the intensity of a wave and probability is directly proportional to the intensity, and also compatible with the experimenatal results explanation does not give a relief to me...

 

[teodorakis]

And one other thing. Should i accept this as a postulate of the qm and do not attempt to look for further deeper explanation. In this case i will accept the probability of a particle is proportional to the intensity of the wave which is defined as the amplitude squared. Or should i interpret it as some joint probability of "meeting" the detector and detected.