16
votes
Accepted
Contradiction in my understanding of wavefunction in finite potential well
Even classically, particles with a fixed total energy spend more time near the turning points since this is where the motion is the slowest. The probably of finding the particle in a small region ...
- 38.4k
11
votes
Contradiction in my understanding of wavefunction in finite potential well
Imagine a perfectly elastic ball dropped vertically onto a flat surface. The ball heads for the point of lowest potential, ie the ground, but because of conservation of energy it bounces back to its ...
- 18.4k
8
votes
Accepted
The reasoning of the definition of $S$-matrix
As lurscher already said correctly, the idea is to choose an initial and a final time that is (in time) away from the actual time-dependent interaction in order to make sure that the interaction does ...
- 6,328
4
votes
Accepted
Wrong formuation of Heisenberg uncertainty principle
Don't be confused it is definitely $$\Delta E \Delta t \geq \frac{\hbar}{2}$$
Your professor typo'd here, as if what he wrote then we could measure energy with any arbitrarily small error we want ...
4
votes
In Quantum Mechanics is it possible to apply time evolution operator to wavefunction?
Using kets, because the evolution operator is linear, we can write :
$$|\psi(t) \rangle = \frac{1}{\sqrt 2}\left(e^{-iHt/\hbar} |\psi_1\rangle + e^{-iHt/\hbar}|\psi_2\rangle\right) = \frac{1}{\sqrt{2}}...
- 3,544
3
votes
Physical meaning of adding two spin operators
There are two ways of understanding the question.
Given the unit vector $\hat n=(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta)$, the general linear combination $\hat S_{\hat n}=\hat n\cdot \vec S= ...
- 38.4k
3
votes
Accepted
Spherical Bessel Equation has different forms?
To arrive at this, I begin manipulating the Radial equation:
$$\frac{d}{dr}\left(r^2\frac{dR}{dr}\right)-\frac{2mr^2}{\hbar^2}(V(r)-E)R=l(l+1)R$$
You forgot some factors of $r$ while manipulating ...
- 111
3
votes
Contradiction in my understanding of wavefunction in finite potential well
You are probably looking at an energy eigenfunction, a wavefunction which has a definite energy. The statement "$E-V$ is the kinetic energy" does not apply to a single position but to the ...
3
votes
Accepted
Question on computation with the bra-ket-notation
For an interval $\Delta$ we can define $P(\Delta)$ in the position representation (using the bra-ket notation) as follows:
$$ \langle x|P(\Delta) := \mathbf 1_{\Delta}(x) \, \langle x| \quad, \tag{1}$$...
- 3,562
3
votes
Accepted
In Quantum Mechanics is it possible to apply time evolution operator to wavefunction?
This is correct. You can easily derive it from bra-ket notation by projection unto a position state, let
$$
|\psi(t_0) \rangle = c_1|\psi_1\rangle + c_2 |\psi_2\rangle
$$
We have $\hat U(t)|\psi_n\...
- 1,569
3
votes
Accepted
How to second-quantize an operator if the field operator is a spinor
Let $\mathfrak h $ be the $1$-particle Hilbert space. Let $\mathcal O$ be an operator on $\mathfrak h$. The second-quantized version of this operator acts on an $k$-particle state $S_\nu(\phi_1\otimes\...
- 3,544
3
votes
Confusion about the Wigner-Eckart theorem
In principle there exist tensor operators of arbitrary angular momentum/spin - if you consider that the angular momentum generators $L_i$ themselves are a vector operator, then e.g. $L_i$ in a spin-3/...
- 106k
2
votes
Accepted
Why is light with higher frequency more destructive, if it emits the same amount of electrons?
That "high enough" conditional can be a big deal. Bodies are not made of a uniform material with a single response to a single frequency. Ejecting electrons (ionization) is harmful to many ...
- 33.7k
2
votes
Accepted
Berry's phase for an electron in a two-level system
The problem is that when you use this parametrization of the "spin-up" state, the wavefunction is not single-valued in $\theta$. Namely, $|\theta+2\pi,\phi\rangle=-|\theta,\phi\rangle$. The ...
- 5,749
2
votes
How does string theory relate superposition and general relativity?
Theories that allow for superpositions of space being curved in different ways have all of their inconsistencies show up at high energies. The new degrees of freedom which come to the rescue in string ...
- 4,314
2
votes
Do-It-Yourself physics experiment
The first cyclotron was a tabletop device.
Then again, a cyclotron requires a source of ions, I don't know whether that is doable as a home project.
Also, I don't know how high of a vacuum is required....
- 15.2k
2
votes
Why possibility for X-ray to excite inner electrons higher than outer electrons?
I was wondering this myself! I found part of the answer and thought I'd share it here for others.
Yes, what the original poster had in mind is correct. The inner core electrons have a much higher ...
- 21
2
votes
Criterion for stationary density matrix
Choose a basis in which $H$ is diagonal and assume for its eigenvalues $H_i\neq 0$. Then $C:= [\rho,H]$ has matrix elements
$$ C_{ij} = (H_i - H_j)\rho_{ij},$$
and so
\begin{align}\mathrm{tr}([\rho,H]^...
- 106k
2
votes
Does passing an electric current along a strip of metal submerged in saltwater cause anything?
Yes. Pumping current into the hull of a ship in moored storage has been used for decades to prevent the hull from corroding in constant contact with sea water. To accomplish this, a very large carbon ...
- 73k
2
votes
Accepted
Why are hidden variables hidden?
Answer to question in 1st paragraph is "no".
The term "hidden" here is not being used to signify "impossible to detect"; it is being used to signify "a physical ...
- 46.2k
2
votes
Accepted
Probabilities of eigenfunctions
15+5+3=23. Probability must add to one.
Note added: The ratios are
$$
1+ \frac 13 + \frac 1 5 = \frac{15+ 5+ 3}{15}= \frac {23}{15}
$$
- 41.3k
1
vote
Confused about the scattering Operator in LSZ reduction formula
In Greiner's Field quantization book, Chapter 9 on the LSZ reduction formalism, he states $$S_{fi}=\langle q_1,...,q_m;\text{out}| p_1,...,p_m;\text{in}\rangle\tag{9.10}$$
This might be written ...
- 111
1
vote
How does string theory relate superposition and general relativity?
So I know that in general relativity, superposition cannot be true.
General Relativity is a classical theory, and is not quantized yet, its quantization is a matter of current research, and string ...
- 221k
1
vote
Accepted
Separability of an Hamiltonian with spin
Yes, $[a,L_j]=0,[a,L^2]=L_j[a,L_j]+[a,L_j]L_j=0$ if $a$ is the scalar under rotation.
- 26
1
vote
Accepted
Criterion for stationary density matrix
Assume that the (complex) Hilbert space $H$ is finite-dimensional. For two hermitian operators $A$, $B$ on $H$ it holds that
$$[A,B]^\dagger = - [A,B]$$ and thus
$$\forall \psi \in H:\, \left(\psi,[A,...
- 3,562
1
vote
Accepted
Deriving a recursion relation for Clebsch-Gordan coefficients
Where you went wrong is comparing apples and oranges. Your superb text is right.
By the time it utilized (3.369), it "translated" the unconventional (3.370) into the standard convention, (3....
- 47.4k
1
vote
Accepted
Can the expectation value behave like a function?
Consider a harmonic oscillator with mass $m$ and frequency $\omega$,
and the state :
$$|\psi\rangle = \frac 1{\sqrt{2}}\left(|0\rangle + i|1\rangle\right)$$
In this state we have :
\begin{align}
\...
- 3,544
1
vote
Nonvanishing expectation value lesser Green's function
In condensed matter many-body physics vacuum usually means a state filled up to the Fermi energy (such as the ground state of conduction band in a metal), i.e.,
$$
a_\mathbf{p}|0\rangle = 0 \text{ if ...
- 37.4k
1
vote
Accepted
Pure state vs mixed state in this example
Yes your reasoning is entirely correct. You would now describe the state at hand with the density operator
$$ \rho = |c_1|^2 |\psi_1 \rangle \langle \psi_1| + |c_2|^2 |\psi_2 \rangle \langle \psi_2|$$
...
- 877
1
vote
Accepted
Question regarding exercise of dynamics of spin-1/2 system
You need to do everything explicitly in the basis $\{|+\rangle,|-\rangle\}$.
So you have
$$|\psi(t)\rangle=\begin{pmatrix}
\psi_+(t) \\ \psi_-(t)
\end{pmatrix}
\quad\text{and}\quad
|\tilde{\psi}(t)\...
- 25.9k
Only top scored, non community-wiki answers of a minimum length are eligible