Linked Questions

75
votes
3answers
11k views

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 ...
11
votes
1answer
1k views

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, $$u(x) = \int dx' G(x;x')\rho(x'),$$ ...
18
votes
2answers
9k views

How exactly is the propagator a Green's function for the Schrodinger equation

Sakurai mentions that the propagator is a Green's function for the Schrodinger equation because it solves $$\left(H-i\hbar\frac{\partial}{\partial t}\right)K(x,t,x_0,t_0) = -i\hbar\delta^3(x-x_0)\...
2
votes
1answer
2k views

Why is the propagator the Green's function for Schrodinger equation? [duplicate]

Sakurai says that the propagator is simply the Green's function for the time-dependent wave equation satisfying $$\left [ -\frac{\hbar^2}{2m} \triangledown ''^2+V(\mathbf{x''})-ih\frac{\partial }{\...
2
votes
1answer
899 views

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 ...
5
votes
1answer
1k views

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 ...
0
votes
1answer
818 views

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-...
5
votes
2answers
210 views

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
votes
1answer
258 views

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 ...
1
vote
1answer
63 views

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 ...