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11 votes

Bound states between neutrinos using Schrödinger's equation?

A quick back-of-the envelope estimate, in the style of Fermi: For a neutrino mass of m ~ 1 eV, and a Planck mass M ~ $10^{27}$ eV, and supplanting the newtonian potential $(m/M)^2/r$ for the Coulomb ...
Cosmas Zachos's user avatar
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 ...
hft's user avatar
  • 21.8k
7 votes
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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 ...
Mateo's user avatar
  • 428
7 votes
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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 ...
Valter Moretti's user avatar
6 votes
Accepted

How to get a lower bound of the ground state energy?

It depends a lot on the specifics of the Hamiltonian and system being examined. One approach that works sometimes is to cast the computation as a semidefinite program and use the duality theory there ...
ors's user avatar
  • 549
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 ...
Emilio Pisanty's user avatar
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 (...
ors's user avatar
  • 549
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 ...
Níckolas Alves's user avatar
4 votes
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Determining the Sign of $E$ When Solving the Time-Independent Schrödinger Equation

Normally, if the potential is finite at infinity, one often set this potential to be $0$: $V(\infty)=0$. In this case, bound states have $E<0$. Thus, for hydrogen, all $E<0$ for bound states. ...
ZeroTheHero's user avatar
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 ...
Ján Lalinský's user avatar
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 ...
Javier's user avatar
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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 ...
alanf's user avatar
  • 8,474
2 votes

Does the Schrodinger Equation yield a unique wave function and density?

I am learning DFT and the Hohenberg Kohn Theorem of Existence. And it says that there is a one-to-one correspondence between the external potential and the density. However the proofs that I have seen ...
hft's user avatar
  • 21.8k
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}(\...
Qmechanic's user avatar
  • 206k
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 ...
Ján Lalinský's user avatar
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.
my2cts's user avatar
  • 25.3k
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 ...
Frederik vom Ende's user avatar
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 ...
SAKhan's user avatar
  • 1,405
1 vote
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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 ...
E. Anikin's user avatar
  • 1,011
1 vote
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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 ...
AccidentalTaylorExpansion's user avatar
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 $c$ ...
hft's user avatar
  • 21.8k
1 vote

Is it possible to derive Schrödinger's equation from Hamilton's equations?

First of all, it should be stressed that the TDSE (1) cannot be derived from classical physics alone, cf. e.g. this Phys.SE post. However, assuming that the classical system at hand can be quantized, ...
Qmechanic's user avatar
  • 206k
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}{...
Agnius Vasiliauskas's user avatar

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