16 votes
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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 ...
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  • 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 ...
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8 votes
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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 ...
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4 votes
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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 ...
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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}}...
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  • 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= ...
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3 votes
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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 ...
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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 ...
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3 votes
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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: $$ \langle x|P(\Delta) := \mathbf 1_{\Delta}(x) \, \langle x| \quad, \tag{1}$$...
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3 votes
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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\...
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  • 1,569
3 votes
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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\...
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  • 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/...
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2 votes
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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 ...
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2 votes
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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 ...
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  • 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 ...
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  • 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....
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  • 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 ...
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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]^...
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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 ...
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2 votes
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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 ...
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2 votes
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Probabilities of eigenfunctions

15+5+3=23. Probability must add to one. Note added: The ratios are $$ 1+ \frac 13 + \frac 1 5 = \frac{15+ 5+ 3}{15}= \frac {23}{15} $$
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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 ...
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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 ...
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  • 221k
1 vote
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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.
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  • 26
1 vote
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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,...
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1 vote
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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....
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1 vote
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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} \...
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  • 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 ...
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1 vote
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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 $$ \rho = |c_1|^2 |\psi_1 \rangle \langle \psi_1| + |c_2|^2 |\psi_2 \rangle \langle \psi_2|$$ ...
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  • 877
1 vote
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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)\...
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