7
votes
Accepted
Time-evolution operator in QFT
Maggiore is assuming there that the field interacts with itself or with other fields in a stationary way. In other words, there is no direct appearance of time in the total Hamiltonian.
This is a ...
7
votes
Accepted
Does this double well potential contradict the fact that there is no degeneracy for one-dimensional bound states?
If you take $V_0$ infinite, the wave function solution of Schrödinger's equation $\psi(x)$ is forced to vanish on the barrier. So it seems reasonable that the solution on either side of the barrier is ...
7
votes
Accepted
Negative kinetic energy on a step potential
I'm having trouble with the explanation of the kinetic energy on the classically forbbiden region on a step potential ($V=0$ for $x<0$, $V=V_0$ for $x>0$ and $E<V_0$).
...
On the classically ...
5
votes
Does this double well potential contradict the fact that there is no degeneracy for one-dimensional bound states?
No, there is no contradiction.
For any finite height of the barrier, the splitting between eigenvalues remains small but nonzero, and the result holds.
If you truly want to think of the barrier as ...
5
votes
Accepted
Is Schrodinger's cat a problem of how we define identity?
Early versions of quantum theory (called the "Copenhagen Interpretation") contained a (subjectively) weird thing called the "Heisenberg cut" the idea of the Heisenberg cut is (...
4
votes
I need to find the state of the system at a general time, knowing the Hamiltonian and the state at $t=0$
For this particular initial state, you do not need to diagonalize the Hamiltonian. The reason is that this particular initial state is an eigenstate of the Hamiltonian, and as a consequence you can ...
3
votes
Is Schrodinger's cat a problem of how we define identity?
The idea that the property of being alive is emergent doesn't mean it is an illusion. There is a real objective difference between a cat being alive and a cat being dead. A living cat breathes in ...
3
votes
Negative kinetic energy on a step potential
Negative kinetic energy is absurd, right? What's wrong with this calculation?
Kinetic energy value we assign to the particle, based on measurement or the psi function, cannot be negative. When you ...
3
votes
Accepted
Time derivative of complex conjugate wave function
Is $\Psi^*$ a wavefunction? Depends on what you mean by that: $\Psi^*(x,t)$ is certainly a function, it has the property that $\int dx\, |\Psi^*|^2 = 1$ for all $t$ (assuming that $\Psi$ does), so you ...
2
votes
Derivation of Schrödinger equation in Feynman-Hibbs
Briefly speaking, it follows from dimensional analysis that higher-order terms ${\cal O}(\eta^{n\geq 3})$ will [after the Gaussian $\eta$-integration (4.5)] only produce higher-orders terms ${\cal O}(\...
2
votes
Accepted
Quantum Mechanical Current Normalisation
Well, the Schroedinger equation for $\psi$ is always kind of the same; the potential energy or the number of kinetic terms changes, but otherwise the form
$$
\hat{H}\psi = E\psi
$$
remains. But the ...
1
vote
What, exactly, does Schrödinger's wave equation describe (just in plain English, without any of the math please)
Unfortunately, in a specific sense your description is quite wrong.
Many of the aspects that you describe are interpretations that arose years after the framework was first published.
You are ...
1
vote
Accepted
Calculating the expectation value of the angular momentum operator
How do come up with
$$
\frac{1}{\pi}\frac{\hbar}{i} \frac{1}{2\phi}\sin^2{2\phi \pi}?
$$
The answer for the integral must be a number, not a function of $\phi$. The integral
$$
\int_0^{2 \pi} \cos\...
1
vote
Quantum Mechanical Current Normalisation
You just need to normalise your solution to unity, after correcting your expressions. The meaning of this is known as the Born rule.
1
vote
Relationship between unitaries generated by a Hamiltonian and its negative sign
Short answer: Your guess is correct, for general time-dependent Hamiltonians $U_1^\dagger(t)\neq U_2(t)$ and there is usually no relation between these two (we will give two simple counterexamples ...
1
vote
Question on 1D Scattering Resonances
In the theory of scattering one defines phase shift $\delta$ as the change in phase of the outgoing to incoming wave. At those particular points where a resonance occurs this quantity is $\pi$/2. The ...
1
vote
Accepted
Question on 1D Scattering Resonances
Resonances in general are associated with quasi-bound (metastable) states of the scattering potential. When the energy of the falling wave is close to the energy of the metastable state, the ...
1
vote
Accepted
What's the meaning of the momentum operator?
Disperson relation
Before even looking at the Schrödinger eqaution let's look at the wave equation to understand an important concept in physics: the dispersion relation. The wave equation (1D) is ...
1
vote
I need to find the state of the system at a general time, knowing the Hamiltonian and the state at $t=0$
The Hamiltonian for a certain three-level system is represented by the matrix
$$H = \begin{pmatrix}a & 0 & b \\ 0 & c & 0 \\ b & 0 & a\end{pmatrix},$$
where $a$, $b$, and $c$ ...
1
vote
Complex Conjugate of Wave Function's Derivative
Basically it goes like this :
$$\begin{align}
\left( \frac{i\hbar}{2m} \frac{\partial^2 \Psi}{\partial x^2} - \frac{i}{\hbar}V\Psi \right )^* &=\\
\left( \frac{i\hbar}{2m} \frac{\partial^2 \Psi}{...
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