All Questions
Tagged with perturbation-theory quantum-mechanics
21
questions
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proving complex Integral relation from perturbation theory MQ
Someone can help me to prove this identity, it comes from a normalization in MQ. From perturbation theory time dependent we have
$$H(t)=H_0+H’(t)$$
$$|Ψ>=c_a(t)e^{-iE_at/\hbar}|Ψ_a>+c_b(t)e^{-...
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Book suggestions for Perturbation Theory in Quantum Mechanics
I've been searching the web for rigorous books on Perturbation Theory, specifically as an undergraduate physics student. In my experience, many quantum mechanics books lack rigor in their explanations....
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On estimating $\exp(-iHt)$ when $H$ is perturbed
Let us assume that we have an unbounded Hamiltonian $H$ and we perturb it a bit to be $H'=H+ \varepsilon A$. I am sure that estimating $||\exp(-iHt) - \exp(-iH't)||$ belongs to the subject of ...
- 634
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If $\int \bar \Psi \frac{1}{ r_1 - r_2} \Psi d\tau= \frac{20\pi k}{a^5}$, what is $\int \bar \Psi \left( \frac{1}{r_1 - r_2} -E_1 \right) \Psi d\tau$?
Suppose that
$$\Psi = ke^{ar_1}e^{ar_2}$$
and it is known that
$$ \left(E_0 + \frac{1}{\lvert r_1 - r_2 \rvert}\right)\Psi = E\Psi$$
for some constants $E, E_0$.
Assuming that $E$ can be expressed as ...
- 802
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2
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How to integrate $\int \frac{e^{-x}e^{-y}}{\lvert \vec x - \vec y \rvert} dx dy$ for the helium atom?
What methods are used to integrate
$$\int \int \frac{e^{-x}e^{-y}}{\lvert \vec x - \vec y \rvert} dx dy$$
which comes up in perturbation theory calculations such as for the helium atom (8.2.6). The ...
- 802
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1
answer
48
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Evaluation of Integral
How do I evaluate the integral $\int_{0}^{\infty}e^{-t/\tau} e^{i\omega t }dt$
Where $\tau$ is a constant
Akshai Christy Jacob
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1
answer
134
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Schrödinger operator whose potential analytically depends on a parameter – how does the spectrum change?
Let's say we have a self-adjoint operator $H_s$ on the Hilbert space $L^2(\Omega \subseteq \mathbb{R})$ defined by
$$
H_s \, \psi(x) := -\psi''(x) + V_s(x) \, \psi(x) \: ,
$$
where $s \in \mathbb{R}$...
- 651
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202
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Perturbation theory in math and quantum mechanics
This question is about my lets say lack of understanding. I can not make the connection between what i study and what i saw in lectures.
For example in our QM lectures we saw perturbation theory like ...
- 619
1
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1
answer
277
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Expanding a PDE in powers of a small parameter?
I'm working on an assignment for my quantum mechanics class and I've arrived at a nonlinear inhomogeneous partial differential equation for a complex function $S:\mathbb{R}^2\to\mathbb{C}~;~S:(x,t)\...
- 12.5k
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answer
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Why is $\frac{1}{\pi ^{1/4} \sqrt{a}}e^{-x^2/(2a^2)}$ not a gaussian function?
In an quantum mechanics exercise, we were asked to find the ground-state wavefunction of a perturbed harmonic system. The resulting wave-function is $$\psi_0(x) = \frac{1}{\pi ^{1/4} \sqrt{a}}e^{-x^2/(...
user446410
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answer
157
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Complementary Projective operator in time-independent non degenerate perturbation theory
I have some difficulties when I am reading time-independent perturbation theory of Sakurai's book "Modern Quantum Mechanics" p.290. Let me introduce the background of this question first. Originally, ...
- 203
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Quantum mechanics: First order perturbation to eigenstates
On page 254 of "Introduction to Quantum Mechanics Second Edition", by David J. Griffiths he writes the first order correction $\delta u_n(x)$ to the eigenstates $u_n(x)$ as
$$\delta u_n(x)=\sum_{m\ne ...
- 8,528
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Perturbation of eigenvalues of Schrodinger Operator.
Suppose I have a sequence of Schrodinger opertors
$$
T_n=-\Delta +V_n
$$
acting on (a subdomain of) $L^2(\mathbb{R}^d)$. Suppose that I view them as perturbations of the operator
$$
T=-\Delta+V
$$
...
- 1,035
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votes
1
answer
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Is there a book on the purely mathematical version of perturbation theory?
Is there a book on the purely mathematical version of perturbation theory, or all current references just in relation to applied fields like statistics and quantum mechanics? I remember first coming ...
- 131
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perturbation theory $\&$ integrability
Suppose we are studying square-integrable eigenfunctions of a linear operator (e.g. an ordinary differential operator), doing perturbation theory in a small parameter.
Suppose first-order ...
- 351