Quantcast
  • 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.

News

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

Attributions

(propose a free ad)

Site Statistics

145 submissions , 122 unreviewed
3,930 questions , 1,398 unanswered
4,852 answers , 20,624 comments
1,470 users with positive rep
501 active unimported users
More ...

quantum mechanics current operators

+ 4 like - 0 dislike
26 views

How to derive the charge current and the energy current operators in second quantized form in Quantum mechanics ? Also if you could comment in a similar way on the entropy current operator, that will be nice.

The way I am proceeding is simply taking the time derivative of the number operator and the Hamiltonian operator. What is wrong in this approach and what is a more appropriate approach ?

Consider a non-relativistic fermionic/bosonic Hamiltonian given in second-quantized form. For example, we can take the Hubbard Hamiltonian. But please start with a general Hamiltonian H to explain the process in general.

This post imported from StackExchange Physics at 2015-06-15 19:45 (UTC), posted by SE-user cleanplay
asked Jul 5, 2013 in Theoretical Physics by cleanplay (80 points) [ no revision ]
Would you mind including some physical context? About which charge and energy currents are you asking? Is this a non-relativistic setup about which you're asking?

This post imported from StackExchange Physics at 2015-06-15 19:45 (UTC), posted by SE-user joshphysics
@joshphysics I have edited the question to make it more clear.

This post imported from StackExchange Physics at 2015-06-15 19:45 (UTC), posted by SE-user cleanplay

1 Answer

+ 2 like - 0 dislike

Whether you use second quantization formalism or whether you are even talking about classical or quantum systems current is defined via the continuity equation for some quantity, $\hat{O}$,

$$\frac{\partial \hat{O}}{\partial t} + \nabla \cdot \mathbf{\hat{J}} = 0,$$

where I have used hat to denote we are talking about quantum mechanical observables.

The question is can we find a pair of observables for which the above equation holds. For integrable models, such as the Hubbard model, Heisenberg spin chain model, free fermions the answer is yes. We can identify local conserved charges for which the above equation holds.

Now, in the Heisenberg picture we have, $$\frac{d}{dt}\hat{O}(t)=\frac{i}{\hbar}[H,\hat{O}(t)]$$

So if you have some Hamiltonian and some corresponding local conserved charge you compute it's commutator with the Hamiltonian and use that to find the current operator. For instance, for the Hubbard model,

$$ H = -t \sum_{\langle i,j \rangle,\sigma}( c^{\dagger}_{i,\sigma} c^{}_{j,\sigma}+ h.c.) + U \sum_{i=1}^{N} n_{i\uparrow} n_{i\downarrow}$$

the number density current at site $i$, $n_i=c^\dagger_i c_i$ can be easily found by a discretized version of the continuity equation, $$\frac{i}{\hbar}[H,n_i(t)]=- (J_{i+1}-J_{i}) = -t(-i c^\dagger_{i+1} c_i+ h.c.)+t(-i c^\dagger_{i} c_{i-1}+ h.c.),$$ which allows us to identify, $$J_i=t(-i c^\dagger_{i} c_{i-1}+ h.c.) $$ as the particle number current density at site $i$. Please note that the total (sum for all sites) particle number density commutes with the Hamiltonian, assuming periodic boundary conditions. This is very important for integrability.

For more examples (e.g., energy current in the Hubbard model), see this paper: http://arxiv.org/abs/cond-mat/9611007

This post imported from StackExchange Physics at 2015-06-15 19:45 (UTC), posted by SE-user Bubble
answered Jul 13, 2013 by Bubble (210 points) [ no revision ]
Nice answer, thanks for the very useful paper. I found equivalent approach which defines the current in terms of discretized form of polarization.

This post imported from StackExchange Physics at 2015-06-15 19:45 (UTC), posted by SE-user cleanplay

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:
p$\hbar$ysic$\varnothing$Overflow
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).
To avoid this verification in future, please log in or register.




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

Your rights
...