5
votes
Accepted
How can we ignore the diverging term $e^\infty$ in the integral?
You can rewrite the indefinite integral as
\begin{equation}
e^{-a x} f(x)
\end{equation}
where $f(x)$ does not grow exponentially with $x$. (In fact $f(x) \sim e^{\pm i k x}$ is an oscillating ...
- 33k
5
votes
Energy-Momentum tensor in the non-relativistic limit of Klein-Gordon Field
We can ignore the cross-terms in the Klein-Gordon (KG) action from OP's expansion
$$\Phi(\vec{x},t)~=~\frac{\hbar}{\sqrt{2m}}\left(\exp\left(-\frac{imc^2t}{\hbar}\right)\psi(\vec{x},t)
+\exp\left(\...
- 167k
5
votes
Are the shapes of atomic orbitals direct consequence of the Schrödinger equation?
The shape of the orbitals are derived by the Schroedinger equation, more precisely if you look at the hydrogen atom you can separate the variables,
the angular part actually determines the angular ...
- 169
4
votes
The Schrödinger-Equation. A linear differential equation of what order?
No. You are not completely right. The Schrodinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is first order in the derivative ...
- 103
4
votes
Accepted
What does Haag's theorem say about the Schrodinger picture?
Haag's theorem says that the representation of the CCR of a free field cannot be isomorphic to the representation of the CCR of an interacting field.
So if you start with an interacting field, there's ...
- 105k
2
votes
The Schrödinger-Equation. A linear differential equation of what order?
one cannot generally assign an order to the Schrödinger-Equation
This statement is completely incorrect. If you mean a "Schrodinger equation that describes all systems" and so the order ...
- 24.7k
2
votes
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Time-independent amplitude to go from one point to another in Feynman lectures (free particle)
Feynman's expression $$ G({\bf r})~=~\langle {\bf r}_2 | {\bf r}_1 \rangle~=~\frac{\exp(ikr)}{r}\tag{3.7}$$ is the propagator/Greens function $$(\nabla^2+k^2)G({\bf r})~=~-4\pi \delta^3({\bf r})$$
for ...
- 167k
2
votes
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Stationary Schrödinger Equation in Momentum space
Here, $\hat{V}(p)$ is an operator that should be acting on $\hat{\psi}(t,p)$ and therefore that should be acting on $P(p)$. If $\hat{V}(p)$ simply multiplies the wavefunction by some function of $p$ ...
- 2,564
2
votes
Accepted
Product rule for bras and kets
Typically we define the notation $\langle x|\hat \Omega(t) | x'\rangle \equiv \Omega(x,x',t)$, where $\hat \Omega(t)$ is a (generally time-dependent) abstract operator and $\Omega(x,x',t)$ is an ...
- 50.3k
1
vote
Accepted
Ordinarily continuous function of the wave function
Griffiths is appealing to the semantic meaning: ordinarily=usually.
For OP's other questions, in particular the bootstrap equation (2.127), see also e.g. my related Phys.SE answer here.
- 167k
1
vote
Energy-Momentum tensor in the non-relativistic limit of Klein-Gordon Field
You are mixing the real Klein-Gordon field, which describes a neutral particle with the complex Schrödinger field. That is not consistent.
The correct approach starts with the complex Klein-Gordon ...
- 21.7k
1
vote
Accepted
QM: How do electrons affect each other's potential energy?
I just wanted to reach out and answer this question and give you a little bit of feedback. First of all, there is a LOT going on in this question. The line “I would like to understand all of chemistry”...
- 99
1
vote
Schrödinger equation obtain $ψ(x,t)$ from $ψ(x,0)$
Just plug $\psi(x,t)= e^{i(k x-\omega t)}$ into your translation-invariant wave equation and read off what $\omega(k)$ has to be to satisfy it.
For example the free Schrodinger equation
$$
i\hbar \...
- 41.2k
1
vote
Deriving Schrödinger equation from Ehrenfest Theorem
This step is a mathematical one; in general, if
$$
\int f = \int g,
$$
you can't conclude that $f=g$.
But if
$$
\int f \psi = \int g \psi
$$
for a large class of functions $\psi,$ then you sometimes ...
- 111
1
vote
Accepted
Quantum mechanics of a moving bound state in the lab frame
Actually, your two examples here are subtly different:
Consider a non-relativistic quantum-mechanical problem of two bodies interacting via a confining potential 𝑉(𝐫⃗ ) (say, a hydrogen atom) ...
- 33k
1
vote
Are the shapes of atomic orbitals direct consequence of the Schrödinger equation?
I am trying to understand whether the shapes of the orbitals are inevitable given the standard model.
The atomic electron orbitals of atomic physics come about because the atomic potential is ...
- 6,812
1
vote
Accepted
Time dependent Schrodinger equation through variation principle - questions about derivation
Well, let's see.
The Lagrangian is
$$ L~=~ \frac{\langle \psi | i \partial_t -H |\psi\rangle}{2||\psi ||^2} ~+~ c.c.,\qquad ||\psi ||^2~\equiv~\langle \psi | \psi\rangle. \tag{A}$$
An infinitesimal ...
- 167k
1
vote
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Definition of transmission and reflection coefficients for a particle
In general, $R$ and $T$ can be defined with probability current.
In 1D,
$$j = \frac{\hbar}{ 2mi} ( \psi^* \frac{\partial \psi}{\partial x} - \psi \frac{\partial \psi^*}{\partial x} )$$
In 3D, partial ...
- 193
1
vote
Accepted
Radial Schrödinger equation: from $R_l(r)$ to $u_l(r)$
While it seems like you've resolved your particular issue, this might be a good opportunity to think about how operations work more generally, as you will come to see a lot of operator algebra in ...
- 151
1
vote
Accepted
Feynman path integral in an EM field
In principle the Feynman fudge factor (4-8) (which comes from Gaussian momentum integrations of the Hamiltonian phase space path integral, cf. e.g. this Phys.SE post) should not be affected by the E&...
- 167k
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