So let's say that we have som unitary matrix, ##S##. Let that unitary matrix be the scattering matrix in quantum mechanics or the "S-matrix".
Now we all know that it can be defined in the following way:
$$\psi(x) = Ae^{ipx} + Be^{-ipx}, x<<0$$ and $$ \psi(x) = Ce^{ipx} + De^{-ipx}$$.
Now, A and...
Practically it is said that, given two spin states |u⟩ (up) and |d⟩ (down) - which are the spin measured along the +z and -z semiaxes - such that they are orthogonal ( ⟨u|d⟩ = ⟨d|u⟩ = 0), it is possible to write any other spin states using a linear combination of these two (because they are a...
I am unable to complete the first part of the question. After I plug in the function for psi into the differential equation I am stuck:
$$\frac {d \psi (r)}{dr} = -\frac 1 a_0 \psi (r), \frac d{dr} \biggl(r^2 \frac {d\psi (r)}{dr} \biggr) = -\frac 1 {a_0}\frac d {dr} \bigl[r^2 \psi(r) \bigr] =...
I started and successfully showed that the expectation of X_1 and X_2 are zero. However the expectation value of X1^2 and X2^2 which I am getting is <X1^2> = 0.25 + \alpha^2 and <X2^2> = 0.25.
How do I derive the given equations?
The difference between light and very heavy atoms reflects itself in these two schemes.
My question is why one scheme for the vector sum is necessarily the right & suitable sum model for one case, and the 2nd scheme suits the 2nd case ?
In other words, why & how the relative magnitude of the...
I want to exactly diagonalize the following Hamiltonian for ##10## number of sites and ##5## number of spinless fermions
$$H = -t\sum_i^{L-1} \big[c_i^\dagger c_{i+1} - c_i c_{i+1}^\dagger\big] + V\sum_i^{L-1} n_i n_{i+1}$$
here ##L## is total number of sites, creation (##c^\dagger##) and...
Dirac wanted to fix the problems with the Klein-Gordon equation by seeking a new solution to it.
He wanted a relativistic solution so it makes sense that the solution needed to satisfy Einstein's energy-momentum relation.
But why did it need to be of first order in time- and...
In the last paragraph of these notes, https://ocw.mit.edu/courses/physics/8-04-quantum-physics-i-spring-2016/lecture-notes/MIT8_04S16_LecNotes3.pdf, it says how a state with large number of photons is not classical. Why is that? I thought quantum mechanics' laws were most applicable when we...
In everyday language, beauty is an emotional concept. How would you mix that with quantum physics and the mathematics behind it? Or is what he refers to as "beauty" is more like simplicity? I mean, I agree with the man, the Standard Model feels redundantly complex, but I feel like there is...
I'm currently in my first QM course and we have griffiths as our textbook. Although Griffiths is a great read I'm having trouble really conceptualizing QM ideas. We've only covered infinite potential well and wave packets as of now. Are there any additional resources or textbooks in which they...
Homework Statement
Homework Equations
The Attempt at a Solution
Hi All,
My problem is that when I calculate this integral or use software to do it for me I get (3*i*pi)/16, when I've been told that the answer is 1/2i giving a probability of 1/4. Would someone be able to point out where...
If we have a particle, say, an electron and we shoot it straight through an empty box. This box is surrounded by light sources on its two sides:
So, if you consider the above cube, if we shoot a particle in a straight line such that it crosses the face ABEF and it crosses the face HGDC through...
Hello,
can anybody help me with solving the problem.
Homework Statement :[/B]
Calculate minimum uncertainty?
A horizontal beam of laser light of wavelength 604 nm passes through a narrow slit that has width 0.0600 mm . The intensity of the light is measured on a vertical screen that is 2.40...
Hello everyone, I'm an Undergrad and I'm trying to set the Quantum key distribution experiment by using weak laser pulses(the laser diode with attenuated filter). but I don't know how to calculate the photon number per pulse. Can anyone tell me how to calculate the photon number per pulse...
Homework Statement
I am supposed to construct a controlled Hadamard gate
using only single qubit and CNOT gates.
Homework Equations
[/B]
We know that any arbitrary unitary Operator U can be written as the Martrix product U=AXBXC, where X is the NOT-Matrix and ABC=1 (identity matrix)
I've...
Given that position and momentum are Fourier conjugates, what is the derivation for the equation ##\hbar \vec{k} = m \vec{v}##, where momentum-space momentum is defined as ##\hbar \vec{k}## and position-space momentum is defined as ##m \vec{v}##?
Dear All,
I have a couple questions on the double slit experiment I hope you can help shed some light (or photons) on. =)
Arrival Timing of Photons
In a normal double-slit experiment like the above setup, do photons always arrive at the detector at a constant speed (basically, speed of light)...
Homework Statement
Consider Compton scattering of a photon by a moving electron. Before the collision
the photon has wavelength λ and is moving in the +x-direction, and the electron is
moving in the −x-direction with total energy E (including its rest energy mc2). The
photon and the electron...
https://en.wikipedia.org/wiki/Future_of_an_expanding_universe "Over an infinite time there could be a spontaneous entropy decrease, by a Poincaré recurrence or through thermal fluctuations (see also fluctuation theorem)"
Homework Statement
I have ##H'=ax^3+bx^4##, and wish to find the general perturbed wave-functions.
Homework Equations
First-order correction to the wave-function is given by, $$\psi_n^{(1)}=\Sigma_{m\neq n}\frac{\langle\psi_m^{(0)}|H'|\psi_n^{(0)}\rangle}{n-m}|\psi_m^{(0)}\rangle.$$
The...
Homework Statement
Simple question we have to answer:
(Physics) How do I release the electrons from the cathode with a color filter? (The so called
Photoelectric effect)
Homework Equations
none
The Attempt at a Solution[/B]
Here we have a conflict , the so called wave-particle duality...
Hey, I've being studying the Photoelectric effect, I think I understand it superficially. One thing that has been bugging me is this:
"Given that it is possible to move electrons with light and given that the energy in a beam of light is related to its intensity, classical physics would predict...
It seems to me that we don't measure a particle because a particle doesn't have an objective existence independent of the wave function or does it? The wave function in this case would have to be real because you can't have probability without the underlying possibility of a specific outcome...
Given:
##\textbf{E}=\hbar \textbf{k}##
where ##\textbf{k} = [\vec{k}_1, \vec{k}_2,\vec{k}_3, i c \omega]##
If ##\textbf{k}## can vary continuously, how does the equation imply that energy is quantized?
For example, ##y = m x +b## where ##m = \hbar## does not imply quantized ##y##.
For...
In a lecture from a course in QM, it was mentioned that Shroedinger's equation is deterministic in one and two dimensions. But in third dimension it gives unstable solutions, loosing it's determinism.
It was mentioned that "in space of D dimensions Gauss theorem leads to the conclusion that...
I am still in secondary school so I probably shouldn't think about things this complicated (at least that's what it seems to me, complicated), but please correct me if I'm wrong. If I recall correctly, the position of an electron is never certain, and always based on probability, unless...
Can any expert help me in explaining how this example below get the reduced density matrix from the density matrix in bipartite system.
$$\rho =\frac{1}{4}\begin{pmatrix} 1 & 1 & cos(\frac{\alpha}{2})-sin(\frac{\alpha}{2}) & cos(\frac{\alpha}{2})+sin(\frac{\alpha}{2}) \\ 1 & 1 &...
Homework Statement
A beam of electrons with kinetic energy 50 eV is fired at a 100 nm thick crystalline nickel foil.
(a) Calculate the speed of the electrons in the beam.
(b) Based on your answer to part (a), state, with justification, whether classical (Newtonian) mechanics correctly...