• Register
PhysicsOverflow is a next-generation academic platform for physicists and astronomers, including a community peer review system and a postgraduate-level discussion forum analogous to MathOverflow.

Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.


PO is now at the Physics Department of Bielefeld University!

New printer friendly PO pages!

Migration to Bielefeld University was successful!

Please vote for this year's PhysicsOverflow ads!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

Tools for paper authors

Submit paper
Claim Paper Authorship

Tools for SE users

Search User
Reclaim SE Account
Request Account Merger
Nativise imported posts
Claim post (deleted users)
Import SE post

Users whose questions have been imported from Physics Stack Exchange, Theoretical Physics Stack Exchange, or any other Stack Exchange site are kindly requested to reclaim their account and not to register as a new user.

Public \(\beta\) tools

Report a bug with a feature
Request a new functionality
404 page design
Send feedback


(propose a free ad)

Site Statistics

205 submissions , 163 unreviewed
5,054 questions , 2,207 unanswered
5,347 answers , 22,728 comments
1,470 users with positive rep
818 active unimported users
More ...

  Quantum mechanics as classical field theory

+ 9 like - 0 dislike

Can we view the normal, non-relativistic quantum mechanics as a classical fields?

I know, that one can derive the Schrdinger equation from the Lagrangian density

$${\cal L} ~=~ \frac{i\hbar}{2} (\psi^* \dot\psi - \dot\psi^* \psi) - \frac{\hbar^2}{2m}\nabla\psi^* \cdot \nabla \psi - V({\rm r},t) \psi^*\psi$$

and the principle of least action. But I also heard, that this is not the true quantum mechanical picture as one has no probabilistic interpretation.

I hope the answer is not to obvious, but the topic is very hard to Google (as I get always results for QFT :)). So any pointers to the literature are welcome.

This post imported from StackExchange Physics at 2014-07-22 05:15 (UCT), posted by SE-user Tobias Diez
asked Feb 20, 2012 in Theoretical Physics by Tobias Diez (90 points) [ no revision ]
These are quantum field theory notes, but they discuss your question damtp.cam.ac.uk/user/tong/qft.html.

This post imported from StackExchange Physics at 2014-07-22 05:15 (UCT), posted by SE-user Kyle
The Schroedinger lagrangian defines a non-relativistic quantum field theory, which is equivalent to many-body quantum mechanics. You may ask whether this field theory has a classical limit. It does, under the usual condition (the states have to be highly occupied). This means that you should look to Bosonic states, for example the case that $\psi$ describes an integer spin atom.

This post imported from StackExchange Physics at 2014-07-22 05:15 (UCT), posted by SE-user Thomas

5 Answers

+ 7 like - 0 dislike

You certainly couldn't recover quantum effects with a classical treatment of that Lagrangian. If you wanted to recover quantum mechanics from the field Lagrangian you've written, you could either restrict your focus to the single particle sector of Fock space or consider a worldline treatment. To read more about the latter, look up Siegel's online QFT book "Fields" [hep-th/9912205] or Strassler's "Field Theory without Feynman Diagrams" [hep-ph/9205205] for applications of the technique.

This post imported from StackExchange Physics at 2014-07-22 05:15 (UCT), posted by SE-user josh
answered Feb 20, 2012 by josh (205 points) [ no revision ]
+ 4 like - 0 dislike

Can we view the normal, nonrelativistic quantum mechanics as a classical fields?

Yes, you can view the wave function $\psi(x,t)$ as an ordinary complex-valued field in the spirit of, say, classical electrodynamics. This field describes the quantum mechanics of a single electron, but it is classical in the sense that it's an ordinary function $\psi(x,t) : \mathbb R^3\times\mathbb R \to \mathbb C$.

As usual, you can have a lot of fun with the Lagrangian. For instance, you can note that there it has a (global) $U(1)$ symmetry $\psi \mapsto e^{i}\psi$ and apply the Noether theorem. You will find a continuity equation for the quantity $\psi^*\psi$, which we commonly interpret as the probability density.

Of course, the Schrdinger equation is limited to non-relativistic physics, so people started to look for a relativistic equivalent. Dirac's eponymous equation was intended to be precisely that: an equation for a classical field that describes a quantum mechanical electron in a Lorentz-covariant way. Of course, there should be an equivalent of the probability density $\psi^*\psi$, which is always positive, but no matter how you spin it, this just didn't work out, even for the Dirac equation.

The solution to this problem is that electrons don't live in isolation, they are identical particles and linked together via the Pauli exclusion principle. Dirac could only make sense of his equation by considering a variable number of electrons. This is where the classical field $\psi$ has to be promoted to a quantum field $\Psi$, a process known as second quantization. ("First quantization" refers to the fact that the classical field $\psi$ already describes a quantum mechanical particle.)

It turns out that second quantization is also necessary to explain certain corrections to the ordinary Schrdinger equation. In this light, the classical field $\psi$ is really an approximation as it does neglect the influence of a variable number of particles.

The process of considering a variable number of particles is actually quite neat. If you go from one to two particles, you would have to consider a classical field $\psi(x_1,x_2)$ that depends on two variables, the particle positions. Going to $N$ particles, you would have a field $\psi(x_1,\dots,x_N)$ depending on that many variables. You can get all particles at once by considering an operator valued field $\Psi(x)$ instead, which creates a particle at position $x$. It turns out that you can just replace $\psi$ by $\Psi$ in the Lagrangian to get the right equations of motion for all particles at once.

Alas, I have to stop here, further details on second quantization and quantum field theory are beyond the scope of this answer.

Concerning literature, I found Sakurai's Advanced Quantum Mechanics to be a very clear if somewhat long-winded introduction to quantum field theory that starts where the Schrdinger equation left off.

This post imported from StackExchange Physics at 2014-07-22 05:15 (UCT), posted by SE-user Greg Graviton
answered Feb 28, 2012 by Greg Graviton (775 points) [ no revision ]
You have to say specifically that this fails for two electrons. That's the whole thing that makes quantum mechanics different from classical field theory.

This post imported from StackExchange Physics at 2014-07-22 05:16 (UCT), posted by SE-user Ron Maimon
+ 2 like - 0 dislike

Indeed, the true Lagrangian for the Schrdinger equation takes this from

$${\cal L}=i \hbar\psi^*\dot\psi-\frac{\hbar^2}{2m}|\nabla\psi|^2-V({\bf x},t)\psi^*\psi$$

and the action becomes

$$S=\int dtd^3x{\cal L}.$$

A Lagrangian for the Schrdinger equation has a meaning only in a quantum field theory context when you do a second quantization on the Schrdinger wavefunction. This applies in a lot fields and mostly in condensed matter and, generally speaking, to many-body physics. In this case, you have to generalize this equation to the Pauli equation and work with spinor and anticommutation rules to describe ordinary matter.

Then, the probabilistic interpretation applies to the states in the Fock space for the many-body problem you are considering.

This post imported from StackExchange Physics at 2014-07-22 05:15 (UCT), posted by SE-user Jon
answered Feb 20, 2012 by JonLester (345 points) [ no revision ]
+ 2 like - 1 dislike

I am no expert in classical fields, but I guess you have no entanglement there, that is, there is no difference with a single particle, but there is a big difference with more than one.

This post imported from StackExchange Physics at 2014-07-22 05:15 (UCT), posted by SE-user user667
answered Feb 20, 2012 by user667 (20 points) [ no revision ]
+ 0 like - 0 dislike

Tong's lecture notes cover this in detail. In particular, your lagrangian is easy to obtain from a complex scalar field that obeys the Klein-Gordon equation in the non-relativistic limit. This lagrangian does lead to a conserved Nther current of the same form as that of Schrdinger's, but this does not have the interpretation of conserved probability, because $\psi \psi^*$ is not the probability density for a particle to be in specific position at specified time. In order to obtain QM, you'll need to quantize the field, construct the wave-function as the superposition of single-particle position states, then obtain the hamiltonian and the Schrdinger equation.

This post imported from StackExchange Physics at 2014-07-22 05:16 (UCT), posted by SE-user auxsvr
answered Apr 6, 2014 by auxsvr (30 points) [ no revision ]

Your answer

Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead.
To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL.
Please consult the FAQ for as to how to format your post.
This is the answer box; if you want to write a comment instead, please use the 'add comment' button.
Live preview (may slow down editor)   Preview
Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:
Then drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds).
Please complete the anti-spam verification

user contributions licensed under cc by-sa 3.0 with attribution required

Your rights