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Homework Help: Quantum mechanics transitions in an electromagnetic field

  1. Sep 3, 2017 #1
    1. The problem statement, all variables and given/known data
    This is problem (7.1) from John A. Peacock "Cosmological Physics".

    Show that the first-order perturbation term for quantum mechanics with an electromagnetic field, ##(e/m) \mathbf{A \cdot p}## is proportional to the electric dipole moment. What is the interpretation of the ##A^2## term?

    2. Relevant equations
    We have the hamiltonian
    $$H = \frac{1}{2m} (\mathbf{p}-e\mathbf{A})^2 + e\phi + V.$$

    In the book they adopt the Coulomb gauge, where ##\nabla \cdot \mathbf{A}=0## and they say that to first order
    the perturbation is ##H' = \mathbf{A \cdot p}##. This means that ##\phi=0## also I assume?

    The text also says (where it refers to this problem), that the transition rate is calculated with the element (using the Fermi's golden rule I assume) ##\langle i | \mathbf{A \cdot p} | j\rangle##, and we must show that it is proportional to the dipole moment
    $$\langle i | \mathbf{A \cdot p} | j\rangle \propto \langle i | \mathbf{A \cdot x} | j\rangle .$$

    3. The attempt at a solution
    If for simplicity I take ##V=0##, so that ##|i\rangle = \int d^3x\ e^{ip_j x}\,|x\rangle## and it is an eigenvector of the momentum operator, I get
    $$\langle i | \mathbf{A \cdot p} | j\rangle = \mathbf{A \cdot p_j}\ \delta^3 (\mathbf{p_j}-\mathbf{p_i}).$$
    It seems that because of the delta function there are no transitions, what am I missing?

    Also in this case I don't think that ##\langle i | \mathbf{A \cdot x} | j\rangle ## will be proportional to the above result:
    $$ \langle i | \mathbf{A \cdot x} | j\rangle = \int d^3x\ \int d^3y\ e^{i(p_jx-p_iy)} A \cdot x \;\delta^3(\mathbf{x}-\mathbf{y})$$
    $$ \langle i | \mathbf{A \cdot x} | j\rangle = \int d^3x\ e^{i(p_j-p_i)x} A\cdot x.$$
    I don't think this expression is equal to the delta function one above.
  2. jcsd
  3. Sep 8, 2017 #2
    Thanks for the thread! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post? The more details the better.
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