# What is Perturbation: Definition and 422 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.

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1. ### A Synchronous to Newtonian gauge

In the context of cosmology, you can perturb around the FRW background, conventionally:$$g = a^2(\tau)[(1+2A)d\tau^2 - 2B_a dx^a d\tau -(\delta_{ab} + h_{ab}) dx^a dx^b]$$with ##a,b## being spatial indices only (1,2,3). You can do gauge transformations ##\tilde{x} = x + \xi## of the coordinates...
2. ### Linearization Euler's equation

I'm trying to linearize (first order) the Euler's equation for a small perturbation ##\delta## Starting with ##mna (\frac{\partial}{\partial t} + \frac{\vec{v}}{a} \cdot \nabla ) \vec{u} = - \nabla P - mn \nabla \phi## (1) ##\vec{u} = aH\vec{x(t)} + \vec{v(x,t)}## Where a is the scale factor...
3. ### Time-dependent perturbation theory

So I have the solution here and trying to understand what happened at the beginning of the second row! How did we get the exponential $$e^{i(\omega_m - \omega_0 ) t' }$$ ?
4. ### I Help with some confusions about variational calculus

I had some several questions about variational calculus, but seems like I can't get an answer on math stackexchange. Takes huge time. Hopefully, this topic discussion can help me resolve some of the worries I have. Assume ##y(x)## is a true path and we do perturbation as ##y(x) + \epsilon...
5. ### A charge inside a ring, small oscillation

This is the picture of the problem. I attach my solution. I first used a trick with gauss's law to calculate the radial electric field at first order of r. ( where r is small ) ( we can assume ##small r=\delta r##) I used a cylinder at the center of the ring then i calculated the ##\hat{z}##...
6. ### I Linearising Christoffel symbols

Carroll linearising by perturbation ##g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}## has: (Notes 6.4, Book 7.4) ##\Gamma^{\rho}_{\mu\nu}=\frac{1}{2}g^{\rho\lambda}\left( {\partial_{ \mu}}g_{\nu\lambda}+{\partial_{ \nu}}g_{\lambda\mu}-{\partial_{...

34. ### Perturbation from a quantum harmonic oscillator potential

For the off-diagonal term, it is obvious that (p^2+q^2) returns 0 in the integration (##<m|p^2+q^2|n> = E<m|n> = 0##). However, (pq+qp) seems to give a complicated expression because of the complicated wavefunctions of a quantum harmonic oscillator. I wonder whether there is a good method to...
35. ### A Gauge Invariance of Transverse Traceless Perturbation in Linearized Gravity

In linearized gravity we define the spatial traceless part of our perturbation ##h^{TT}_{ij}##. For some reason this part of the perturbation should be gauge invariant under the transformation $$h^{TT}_{ij} \rightarrow h^{TT}_{ij} - \partial_{i}\xi_{j} - \partial_{j}\xi_{i}$$ Which means that...
36. ### I Confused about perturbation theory

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...

48. ### I Raising/Lowering Metric Indices: Explained

If I have a metric of the form ##g_{\mu \nu} = f_{\mu \nu} + h_{\mu \nu}## where ##f_{\mu \nu}## is the background metric and ##h_{\mu \nu}## the perturbation, how do I raise and lower indices of tensors? For instance, I was told that ##G_{ \ \nu}^{\mu} = f^{\mu \nu '} G_{\nu ' \nu }##. But...
49. ### Selection Rules (Time Dependent Perturbation Theory)

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?
50. ### Applying Selection Rules to Determine Non-Zero Ground State Perturbations

Since E_i=0 for the ground state, and $$E_f=\frac{(\hbar)^2l(l+1)}{2I}$$, $$w_{fi}=\frac{E_f-E_i}{\hbar}=\frac{(\hbar)l(l+1)}{2I}$$. So, $$d_f(\infty)=\frac{i}{\hbar}\int_{-\infty}^{\infty}<f|E_od_z|0>e^{\frac{i\hbar l(l+1)t}{2I}+\frac{t}{\tau}}dt$$ My question is in regards to...