Linked Questions
16 questions linked to/from Differential equation (Greens function) satisfied by the kernel using path integrals
3
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Why is the propagator the Green's function for Schrodinger equation? [duplicate]
Sakurai says (in various editions) that the propagator is simply the Green's function for the time-dependent wave equation satisfying
$$\begin{align}&\left [ -\frac{\hbar^2}{2m} \triangledown ''^2+...
- 233
128
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3
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Differentiating Propagator, Green's function, Correlation function, etc
For the following quantities respectively, could someone write down the common definitions, their meaning, the field of study in which one would typically find these under their actual name, and most ...
- 8,693
19
votes
1
answer
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Dirac Delta in definition of Green function
For a inhomogeneous differential equation of the following form
$$\hat{L}u(x) = \rho(x) ,$$
the general solution may be written in terms of the Green function,
$$u(x) = \int dx' G(x;x')\rho(x'),$$
...
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27
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2
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How exactly is the propagator a Green's function for the Schrodinger equation?
Sakurai mentions (in various editions) that the propagator is a Green's function for the Schrodinger equation because it solves
$$\begin{align}&\left(H-i\hbar\frac{\partial}{\partial t}\right)K(x,...
- 2,010
7
votes
2
answers
634
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Equivalence of second-quantized Schrödinger field and its first-quantized path integral formulation
I was recently learning about the the second quantization of the Schrödinger field, and naturally got interested in how it aligns with the field theoretic path integral. So just as a short ...
- 541
2
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1
answer
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Harmonic Oscillator propagator
I'm reading the book on Quantum Field Theory by Anthony Duncan, and I'm a little lost with something of propagators.
He first define the propagator $K(q_f,T;q_i,0)$ as the amplitude of detecting a ...
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7
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1
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Feynman's derivation of the Schrödinger equation
I'm reading the following article:
Feynman's derivation of the Schrödinger equation
In this article, the autor claims that Feynman derivation of the Schrödinger equation was a key aspect of the ...
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1
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Checking that the propagator for Harmonic Oscillator satisfies Schroedinger Equation [closed]
I have the propagator for the harmonic oscillator.
$$K(x_f,x_0,t)=\sqrt{\frac{m\omega}{2 \pi \hbar \sin{wt}}}\exp\left(\frac{i}{\hbar}\frac{m\omega}{2 \sin{\omega t}}((x_0^2+x_f^2)\cos\omega t-...
- 895
6
votes
2
answers
990
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Is the path integral amplitude a wavefunction?
The probability amplitude for a particle to travel from $\mathbf{x}_i $ to $\mathbf{x}_f$ in a time $t$ is given by the path integral
$$ \langle \mathbf{x}_f | e^{-iHt} |\mathbf{x}_i \rangle = \int \...
- 4,064
4
votes
1
answer
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Deriving the Lagrangian form of the Feynman path integral through Gaussian integration
The hamiltonian form of the path integral for the time evolution of a single particle in one dimension (in non-relativistic quantum mechanics) is:
$$\langle x|\hat U(t_2,t_1)|x'\rangle=\int \mathcal ...
- 3,877
3
votes
1
answer
660
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How to understand the kernel as a transition amplitude?
Consider the time evolution operator $U(t_f, t_i)$ that controls the evolution of a wave function according to $|\psi(t_f \rangle = U(t_f, t_i) | \psi(t_i) \rangle$.
As I understand it, the Born ...
4
votes
1
answer
319
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Mysterious factor 2 in the Schrödinger equation derived from Feynman's Kernel
Feynman's Quantum Mechanics and Path Integral has a vivid physical interpretation of the path integral formalism. But I was stumbled on some mathematical details while following his derivation of the ...
- 137
1
vote
1
answer
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Propagator in Path Integral Quantum Mechanism as Green Function of Schrodinger Equation
I'm studying in Ryder's book of QFT. I'm dealing with QM in the path integral approach and he is trying to prove that the propagator $K(x_f t_f;x_i t_i)$ is the Green function of the Schrodinger (S.) ...
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0
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1
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Green's functions co-incidentally appearing in the path integral of relativistic free particle action
When you compute the path integral of the relativistic free particle action, it's turns out to be the same as the Green's function of a classical field. This co-incidence is huge because it derives, ...
- 6,790
5
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1
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What is the meaning of the function of $x$ and $p$ that you get when you cut open the phase space path integral?
When we "cut" an ordinary path integral, we obtain a state in the position representation. That is, if we fix some initial position $x_i$, then the path integral
$$\int_{x_i}^{x_f}Dx e^{-S}$$...
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