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Hamiltonian In magnetic field

  1. Mar 20, 2005 #1
    I have a question regarding the Hamiltonian in a magnetic field.
    First Hamiltonian with potential V is given by
    Ho = (1/2m)*p^2 + V

    but if a vector potential A is also present then
    H1 = (1/2m)*(p+eA)^2 + V

    there is a way to write H1 interm of Ho
    H1 = exp(-ier.A) Ho exp(ier.A)
    where r is position
    and r.A is dot product of position and vector potential

    Any one knows Why? are these both forms equal and how to convert one into another
  2. jcsd
  3. Mar 20, 2005 #2
    The expressions you wrote are two forms of the same Hamiltonian. The difference consists of p (momentum). In H_0 p is the classical momentum (mv) whereas in H_1 p is the canonical momentum (mv+eA).

    BTW. In H_1 you must have (p-eA)

  4. Mar 20, 2005 #3
    Hey clive, yeah it has to be p-eA for it to be equal,
    but how is the third equation equal to the second.
    Last edited: Mar 21, 2005
  5. Mar 21, 2005 #4


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    That [itex] \hat{U} [/itex] is the potential energy operator.Is there any connection between this operator and the magnetic potential vector operator...([itex] \hat{A}[/itex])?
  6. Mar 21, 2005 #5
    What do you mean by "e" (in the third equation) ziyad?
  7. Mar 21, 2005 #6


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    E sarcina elementara "e".

    It's the elementary charge "e",i'm sure of it.
  8. Mar 21, 2005 #7
    No, this is not the way.


    What you have written down comes from

    eiR∙a/h_bar P e-iR∙a/h_bar = P - a .

    In this relation, however, the vector a is fixed. It has no dependence on space (although, it could depend on time). If it were the case that a had some kind of functional dependence on space – and hence, we would have "±iR∙a(R)/h_bar" in the exponentials – then it would no longer be the case that the left-hand side equals the right-hand side.


    The general expression goes like this:

    eiG(R,t)/h_bar P e-iG(R,t)/h_bar = P – (grad G)(R,t) .


    There is definitely something more to be said here (though, for the moment, I'm not quite sure what it is).
  9. Mar 21, 2005 #8
    clive 'e' is the charge

    ok i wrote it wrong. here i scanned the equations

    eye in the sky, yes ur correct. A is a constant vector.

    what is that equality called. and where can i read more about it.

    looking at the equation u mentioned. by changes signs
    exp(-ier.A/hbar) Ho exp(ier.A/hbar) = Ho+eA

    Now if i expand the square in the original H1 i get

    H1 = Ho + (1/2m)(ep.A + eA.p+ (eA)^2)

    How do i make this second term equal to just eA so i can use the equality u mentioned.

    eye in the sky, how r u able to write equations in this forum.
    Last edited: Mar 22, 2005
  10. Mar 21, 2005 #9


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    Use the Latex code.

  11. Mar 22, 2005 #10
    I don't know of any special names for these equalities. The first relation (in which the vector a is a constant) is directly related to the notion of the operator mR as the generator of "boosts". A similar equality, in which the roles of P and R are reversed, is directly related to the notion of the operator P as the generator of "(spatial) translations". All of these equalities are equivalent to the basic canonical commutation relations for R and P; i.e.

    [Rk, Rl] = [Pk, Pl] = 0 ,

    [Rk, Pl] = ihbar δkl .

    Alternatively, they are equivalent to:

    (i) the action of P in the {|r>} representation is given by -ihbargrad ;


    (ii) the action of R in the {|p>} representation is given by ihbargradp ;

    As for where to read more about these matters, I'm sorry but I can't think of any good elementary references. Perhaps someone else can point you in the right direction.
    You have made an error here. The right-hand side should read H1, as before.
    If A is a constant vector, then there is no reason to include it in the Hamiltonian. More generally, if curl A = 0, then there is no reason to include A in the Hamiltonian at all. In that case, the equations of motion for the system can be derived from a "scalar potential" alone without the need for the introduction of a "vector potential". (This fact has a deep connection with the equality quoted above in which the expression P – (grad G)(R,t) appears on the right-hand side.)

    Having said that, I am beginning to see a little better what I was talking about when I said (back in post #7):
    Yes, yes. There is something here about being able to 'remove' the A(R,t) term by means of a certain kind of unitary transformation if, and only if, curl A = 0 (i.e. A(r,t) = grad G(r,t) , for some function G). ... But it's all still a bit too 'fuzzy' for me to put my finger on.
    If you click on the "quote" tab on any given post, you will be able to see precisely how the equations have been produced. Whenever I need special symbols (e.g. Greek letters), I copy and paste them from a "storehouse" of symbols in a WORD document I keep. Those symbols have been gathered by clicking on INSERT → SYMBOL in the upper menu of Microsoft WORD.

    Perhaps using the Latex code can be easier. I haven't tried.
  12. Mar 22, 2005 #11
    This is getting really complicated :cry:

    a friend told me to look at H1*Phi(r) for arbitrary phi(r)
    where phi(r) is teh wavefunction

    and both forms will give the same result.
    although i'm not sure how exactly
  13. Mar 22, 2005 #12
    We have

    e-ieR∙A/h_bar Ho eieR∙A/h_bar = H1 ,

    where A is a constant vector, and

    Ho = P2/2m + V(R) ,


    H1 = (P + eA)2/2m + V(R) .

    Now what exactly do you want to show?
  14. Mar 22, 2005 #13
    That these both r same

    e-ieR∙A/h_bar Ho eieR∙A/h_bar = H1 ,
    H1 = (P + eA)2/2m + V(R) .
    one can be derived from another, or H1*phi(r) for arbitrary Phi(r) should give same result using either formulas.

    Sorry dude. I'm bugging u so much
  15. Mar 22, 2005 #14
    It sounds like you want to say that Ho and H1 have the same spectrum of eigenvalues. Is that what you want to say?
  16. Mar 22, 2005 #15
    actually both forms of H1 should have the same spectrum of eigenvalues.
  17. Mar 22, 2005 #16
    Okay, now we are "rolling".

    First of all note that the operator

    S(R) ≡ e-ieR∙A/h_bar

    is a unitary operator. That is,

    S-1(R) = S(R) .


    H1 = S(R)HoS(R) .

    Therefore, a ket |ψ> is such that

    Ho|ψ> = E|ψ> ,

    if, and only if, the ket |φ> ≡ S(R)|ψ> is such that

    H1|φ> = E|φ> .

    This shows that Ho and H1 have same spectrum of eigenvalues.

    Next, observe that

    φ(r) ≡ <r|φ>

    = <r|S(R)|ψ>

    = <r|e-ieR∙A/h_bar|ψ>

    = e-ier∙A/h_bar <r|ψ>

    = e-ier∙A/h_bar ψ(r) .

    This shows that corresponding eigenvectors of H1 and Ho (i.e. the ones which have the same eigenvalue) differ by a mere phase factor (given by
    e-ier∙A/h_bar ).

    ... Is there anything else?

    Note that to be completely rigorous, however, we would need to say a few words also about how the above conclusions are still valid in case of degeneracies. (This requires further thought.)
  18. Mar 22, 2005 #17
    :smile: :smile:
    That wasn't so bad. It looks so simple now.

    Eye in the sky. thanks for ur help dude. I think thats the solution, i don't think degeneracy is needed. this is perfect as it stands.

  19. Mar 22, 2005 #18
    So glad to have been of help! :wink:

    (Good luck on the exam!!! :smile: )
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