New answers tagged perturbation-theory
0
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Is there any relevance to these perturbatively defined mode operators in interacting Quantum Field Theories?
Yes the field $\phi_0$ is the standard free field. $\phi_1$ solves $(\Box + m^2 )\phi_1=\phi_0^{3}$ and $\phi_2$ for example is a solution of $(\Box + m^2 )\phi_2=\phi_0^{2}\phi_1$. Graphically $\...
- 334
1
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How to Derive an Expression for First-Order Correction in Adiabatic Perturbation Theory?
Consider the time-dependent Schrodinger equation
$$
i\hbar |\dot\psi_m\rangle = H |\psi_m\rangle.
$$
If the state was an eigenvector of $H$, it would be easy to solve it. The solution would be ...
- 3,349
2
votes
Taking derivative with respect to quantum canonical ensemble expectation value
IMHO the OP is mislead by a recourse to imaginary time... which is not warranted here. Indeed, let us assume that we know the exact eigenstates of $H$:
$$
H|n\rangle = E_n|n\rangle,
$$
(these coule be ...
- 65.1k
4
votes
Taking derivative with respect to quantum canonical ensemble expectation value
On a side note, you can avoid the Trotter formula by using instead the standard interaction picture (if you are already familiar with it from previous QM courses). I will write $\partial_{x_i} = \...
- 15.1k
3
votes
Taking derivative with respect to quantum canonical ensemble expectation value
$$\partial_{x_b} \exp(-\beta H/N)\approx -\exp(-\beta H/N) (\frac{\beta}{N})(\partial_{x_b} H) $$
My question is, what is the strict justification for the last step? How to prove that the error ...
- 23.3k
1
vote
Accepted
Stark Effect for the multiplet $n = 2$ (Gottfried, problem 12, chapter 5, 2nd ed.)
(a) Calculate the matrix elements of $eEz$ in the basis that diagonalizes $H_{fs}$. Show that
the $|m|= 3/2$ states are unaffected,
whereas the perturbed energies and eigenstates in the
space of the $...
- 23.3k
1
vote
Can the non-degenerate perturbation theory formula for higher-order energy corrections be used in case of degenerate perturbation theory?
OP's linked reference (Ref. 1) shows in eq. (1.2.5) that OP's eq. (1.1.26) with $s=1$ may be violated for degenerate (time-independent) perturbation theory. For $s=1$ one needs to impose an additional ...
- 213k
1
vote
Accepted
Is the quadratic stark effect on the ground state of hydrogen not $E_{100}^{(2)}=-\frac{9}{4}a_0^3\mathcal{E}^2$?
I cross posted this question to MSE, where it got an answer (from myself). I realized the issue was that we were ignoring scattering states when calculating the second order perturbation, so we fall ...
- 131
1
vote
Accepted
Spin-Orbit coupling eigenvectors
Consider the first equation
$$𝑐_1(𝑙−𝑚)=\sqrt{𝑙(𝑙+1)−𝑚(𝑚+1)}𝑐_2$$
A solution is $c_1=\sqrt{𝑙(𝑙+1)−𝑚(𝑚+1)}$ and $c_2=l-m$. However, the associated quantum state is not normalized since
$$\...
- 3,708
1
vote
Accepted
Confusion about integral form of interaction potential in Dyson series
In short, it is important (for us all) to distinguish between operators defined in various pictures, and how these definitions relate to each other. P&S define the interaction in equation 4.12, ...
- 3,749
1
vote
Accepted
Baumann's Cosmology: if adiabatic perturbations are generated by shifting conformal time, can't they just be gauged away?
In this context, $\eta \to \eta + \pi(\eta,\mathbf{x})$ shouldn't be viewed as a gauge transformation. If you were to make that transformation in an unperturbed FLRW spacetime, you would end up with ...
- 6,987
3
votes
Integrability of the many-body problem
Statement of problem
One of the key insights of the KAM theorem is that not all modes of the perturbation have the same size. Consider an unperturbed integrable Hamiltonian $H_0(J^A)$, where the ...
- 20.7k
0
votes
Performing renormalization on the interaction constant $g$ to prevent UV divergence in BEC system
OP needs to give a lot more context when asking question. It would be helpful for anyone who attempts to answer the question, and ultimately help to get it answered. I believe your $\epsilon^{(2)}$ ...
- 458
0
votes
Performing renormalization on the interaction constant $g$ to prevent UV divergence in BEC system
Renormalization involves four sets of scales:
cutoff scales, e.g. $\Lambda$ (and/or $\mu$ for IR divergent theories, and possibly other cutoff scales in condensed matter systems with special ...
- 3,749
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