What is Perturbation theory: Definition and 261 Discussions
In mathematics, physics, and chemistry, perturbation theory comprises mathematical methods for finding an approximate solution to a problem, by starting from the exact solution of a related, simpler problem. A critical feature of the technique is a middle step that breaks the problem into "solvable" and "perturbative" parts. In perturbation theory, the solution is expressed as a power series in a small parameter
ϵ
{\displaystyle \epsilon }
. The first term is the known solution to the solvable problem. Successive terms in the series at higher powers of
ϵ
{\displaystyle \epsilon }
usually become smaller. An approximate 'perturbation solution' is obtained by truncating the series, usually by keeping only the first two terms, the solution to the known problem and the 'first order' perturbation correction.
Perturbation theory is used in a wide range of fields, and reaches its most sophisticated and advanced forms in quantum field theory. Perturbation theory (quantum mechanics) describes the use of this method in quantum mechanics. The field in general remains actively and heavily researched across multiple disciplines.
I've arrived to an expected answer, but I am not sure at all that the process was what the problem statement wants.
First, I considered ##0=(t+\delta t)^2-(x+vt)^2-(t^2-x^2) \approx 2t \delta t - 2xvt - v^2t^2##. Ignoring ##O(v^2)## gives ##\delta t=vx##, i.e., ##t \rightarrow t+vx##.
Keeping...
When it comes to scattering in QED it seems only scattering cross sections and decay rates are calculated. Why is that does anyone calculate the actual evolution of the field states or operators themselves like how the particles and fields evolve throughout a scattering process not just...
Hi,
I have to find the eigenvalue (first order) and eigenvector (0 order) for the first and second excited state (degenerate) for a perturbated hamiltonian.
However, I don't see how to find the eigenvectors.
To find the eigenvalues for the first excited state I build this matrix
##...
Hello! I am reading this paper and in deriving equations 6/7 and 11/12 they claim to use second oder time dependent perturbation theory (TDPT) in order to get the correction to the energy levels. Can someone point me towards some reading about that? In the QM textbooks I used, for TDPT they just...
We know that we need to go to 5th order in perturbation theory to match 10 decimals of g-2 for electron, theory vs. experiment. But let us not assume QED is pure and independent, but it's a lower energy limit of GSW (not Green-Schwartz-Witten from superstrings) electroweak theory. Has anyone...
Townsend, quantum mechanics
" In our earlier derivation we assumed that each unperturbed eigenstate ##\left|\varphi_{n}^{(0)}\right\rangle## turns smoothly into the exact eigenstate ##\left|\psi_{n}\right\rangle## as we turn on the perturbing Hamiltonian. However, if there are ##N## states
##...
McIntyre, quantum mechanics,pg360
Suppose states ##\left|2^{(0)}\right\rangle## and ##\left|3^{(0)}\right\rangle## are degenerate eigenstates of unperturbed Hamiltonian ##H##
Author writes:
"The first-order perturbation equation we want to solve is
##...
The transition probability -- the probability that a particle which started out in the state ##\psi_a## will be found, at time ##t##, in the state ##\psi_b## -- is
$$P_{a \to b} = \frac{|V_{ab}|}{\hbar^2} \frac{sin^2[(\omega_0 - \omega)t/2]}{(\omega_0 - \omega^2}.$$
(Griffiths, Introduction...
I tried to use the degenerated perturbation theory but I'm having problems when it comes to diagonalizing the perturbation q1ˆ3q2ˆ3 which I think I need to find the first order correction.
Given the unperturbed Hamiltonian ##H^0## and a small perturbating potential V. We have solved the original problem and have gotten a set of eigenvectors and eigenvalues of ##H^0##, and, say, two are degenerate:
$$ H^0 \ket a = E^0 \ket a$$
$$ H^0 \ket b = E^0 \ket b$$
Let's make them...
Hello folks, I am currently studying from Griffiths' Introduction to Quantum Mechanics and I've got a doubt about good quantum numbers that the text has been unable to solve.
As I understand it, good quantum numbers are the eigenvalues of the eigenvectors of an operator O that remain...
Of course, this question consisted of two parts. In the first part, we needed to calculate the first-order correction. It was easy. In all the books on quantum mechanics I saw, only first-order examples have been solved. So I really do not know how to solve it. Please explain the solution method...
If I plug the solution into the Schrodinger equation I get
$$(i \hbar \partial_t - H)\ket{\psi} = 0$$
Since I know that the zeroth-order expansion is lambda is already a solution I think this is equal to
$$(i \hbar \partial_t - H)e^{i\phi} e^{-i\gamma}\ket{\delta n} = 0$$
If now I carry on with...
In quantum chemistry, the MP rows (MP2, MP3, MP4, etc) can converge both quickly and slowly, and for some cases (e.g. CeI4 molecule) they even diverge instead of converging.
My question is quite philosophic: what is the “mathematical cornerstone”, or “philosophical cornerstone” of the...
I was reading in the Book: Introduction to Quantum Mechanics by David J. Griffiths. In chapter Time-Dependent Perturbation Theory, Section 9.12. I could not understand that why he put the first order correction ca(1)(t)=1 while it equals a constant.
In This wikipedia article is said:
"If the quantum field theory can be accurately described through perturbation theory, then the properties of the vacuum are analogous to the properties of the ground state of a quantum mechanical harmonic oscillator, or more accurately, the ground state of a...
Hello! I saw in many papers people talking about the effects of a time dependent perturbation (usually an oscillating E or B field) on the energy levels of an atom or molecule (for now let's assume this is a 2 level system). Taking about energy makes sense when the hamiltonian is time...
Hello,
I am looking for a reference which describe perturbation theory with two parameters instead of one. So far, I did not find anything on the topic. It might have a specific name and I am using the wrong keywords. Any help is appreciated.
To be clear, I mean I have ##H =...
Hello all, I would like some guidance on how to approach/solve the following QM problem.
My thinking is that Fermi's Golden Rule would be used to find the transitional probability. I write down that the time-dependent wavefunction for the free particle is...
I've been assigned to do a problem from Landau which you can read below:
I have no problem with finding the energy. Then I write down the equations:
\begin{equation*}
\begin{cases}
(V_{11}-E^{(1)})|c_1|e^{i\alpha_1} + V_{21}e^{i\alpha_2}|c_2| = 0\\
V_{12}e^{i\alpha_1}|c_1| +...
Do any of you know of an article or book chapter that discusses the difference between a discontinuous potential well of length ##2L##
##V(x)=\left\{\begin{array}{cc}0, & |x-x_0 |<L\\V_0 & |x-x_0 |\geq L\end{array}\right.##
and a differentiable one
##\displaystyle V(x) = V_0...
Hello! Let's say we have 2 states of fixed parity ##| + \rangle## and ##| - \rangle## with energies ##E_+## and ##E_-## and we have a P-odd perturbing hamiltonian (on top of the original hamiltonian, ##H_0## whose eigenfunctions are the 2 above), ##V_P##. According to 1st order perturbation...
Suppose we have a hamiltonian $$H_0$$ and we know the eigenvectors/values:
$$H_0 |E_i \rangle = E_i|E_i \rangle $$
We then add to it another perturbing Hamiltonian:
$$H’$$ which commutes with $$H_0.$$ According to nondegenerate first order perturbation theory:
$$\langle H \rangle \approx...
Hi,
I just need someone to check if I am on the right track here
Below is a mutual Coulomb potential energy between the electron and proton in a hydrogen atom which is the perturbed system:
##V(x) =
\begin{cases}
- \frac{e^{2}}{4\pi\epsilon_{0}}\frac{b}{r^{2}} \text{for } 0<r \leq 0 \\
-...
I am revising perturbation theory from Griffiths introduction to quantum mechanics.
Griffiths uses power series to represent the perturbation in the system due to small change in the Hamiltonian. But I see no justification for it! Other than the fact that it works.
I searched on the internet a...
Hi everyone,
I'm struggling with the proof for the second order energy correction for perturbation theory when substituting in the first order wavefunction. I have attached an image of my current proof for it below, but I'm not sure whether this is the correct approach for it (the H's in the...
I was learning about Degenerate Perturbation Theory and I encountered the term 'Degenerate Subspace', I didn't really understand what it meant so I came here to ask - what does it mean? will it matter if i'll say 'Degenerate space' instead of 'Degenerate Subspace'? and subspace of what? (...
Looking at. <psi|AB|theta>, under what conditions would this be equal to <psi|A|theta> * <psi|B|theta> I’m just getting into perturbation theory
and am running into confusing notation. Thanks john
Hey there,
This question was asked elsewhere, but I wasn't really satisfied with the answer.
When I learned about Fermi's golden rule, ##{ \Gamma }_{ if }=2\pi { \left| \left< { f }|{ \delta V }|{ i } \right> \right| }^{ 2 }\rho \left( { E }_{ f } \right)##, it was derived from first order...
Intuitively, I'd say that adding a 4-divergence to the Lagrangian should not affect the eqs of motion since the integral of that 4-divergence (of a vector that vanishes at ∞) can be rewritten as a surface term equal to zero, but...
In some theories, the addition of a term that is equal to zero...
I suppose my question is, since X commutes for H, does this mean that the selection rules are $$<n',l',m'|X|n,l,m>=0$$ unless $$l'=l\pm 1$$ and $$m'=m\pm 1$$, as specified in Shankar?
If I calculate ## <\psi^0|\epsilon|\psi^0>## and ## <\psi^0|-\epsilon|\psi^0>## separately and then add, the correction seems to be 0 since ##\epsilon## is a constant perturbation term.
SO how should I approach this? And how the Δ is relevant in this calculation?
I'm getting confused by the perturbation theory aspect of problem 2.2 in this book. We have to show that the energy eigenvalues are given by
$$E_n = \left(n + \frac{1}{2}\right) \hbar \omega + \frac{3\lambda}{4} \left(\frac{\hbar}{m\omega}\right)^2 (2n^2 + 2n + 1)$$
For the Hamiltonian...
I have to solve the equation above. I haven't heard about an exact method so I tried to apply perturbation theory. I don't know much about it so I would like to ask for some help.
First I put an ##\epsilon## in the coefficient of the non-linear ##\xi^2(t)## term:
##\ddot{\xi}(t)=-b\xi...
Hello! In Griffiths chapter on Time independent perturbation theory, he has a problem (9.20) in which he asks us to calculate the first order contribution to the electron Hamiltonian in an atom if one takes into account the magnetic dipole/electric quadrupole excitations, beside the electric...
Homework Statement
Homework Equations
VD= -1/(8m2c2) [pi,[pi,Vc(r)]]
VC(r) = -Ze2/r
Energy shift Δ = <nlm|VD|nlm>
The Attempt at a Solution
I can't figure out how to evaluate the expectation values that result from the Δ equation. When I do out the commutator, I get p2V-2pVp+Vp2. This...
I have been following [this video lecture][1] on how to find gauge invariance when studying the perturbation of the metric.
Something is unclear when we try to find fake vs. real perturbation of the metric.
We use an arbitrary small vector field to have the effect of a chart transition map or...
I'd like to know how to solve the dirac equation with some small gauge potential $\epsilon \gamma^\mu{A}_\mu(x)$ by applying perturbation theory. The equations reads as $$(\gamma^\mu\partial_\mu-m+\epsilon\gamma^\mu A_\mu(x))\psi(x) = 0.$$
The solution up to first order is
$$ \psi(x) =...
Homework Statement
A point mass m hangs at one end of a vertically hung hooke-like spring of force constant k. The other end of the spring is oscillated up and down according to ##z=a\cos(w_1t)##. By treating a as a small quantity, obtain a first-order solution to the motion of m in time...
This isn't explained anywhere so it must be super basic and I'll probably kick myself for not getting it, but on the wiki page for time independent perturbation theory, section 3.1:
https://en.wikipedia.org/wiki/Perturbation_theory_(quantum_mechanics)
It talks about first order corrections and...
Hi everyone,
I am doing a time dependent perturbation theory, in a case when the electron is prepared in a state of the continuous part of the energy spectrum. Existence of the discrete part and the degeneracy of the continuous part is irrelevant at the moment and will not be considered...
Homework Statement
Real atomic nuclei are not point charges, but can be approximated as a spherical distribution with radius ##R##, giving the potential
$$ \phi(r) = \begin{cases}
\frac{Ze}{R}(\frac{3}{2}-\frac{1}{2}\frac{r^2}{R^2}) &\quad r<R\\
\frac{Ze}{r} &\quad r>R \\...
Hello guys,
I'm wondering if there are some important restrctions on the 'applicability' of first order perturbation theory.
I know there's a way to deduce Schwarzschild's solution to Einstein's field equations that assummes one can decompose the 4D metric ##g_{\mu\nu}## as Minkowski...
Hello! I am reading Griffiths and I reached the Degenerate Time Independent Perturbation Theory. When calculating the first correction to the energy, he talks about "good" states, which are the orthogonal degenerate states to which the system returns, once the perturbation is gone. I understand...
What is the nonperturbative approach to quantum mechanics as opposed to perturbative one? When does the latter method fail and one has to apply nonperturbative approach? Please keep your discussion confined within non-relativistic quantum mechanics.
Homework Statement
The photon is normally assumed to have zero rest mass. If the photon did have a tiny mass, this would alter the potential energy the electron feels in the hydrogen atom (due to the Coulomb interaction with the proton). The potential then becomes yukawa potential...