Tagged Questions
2
votes
1answer
87 views
Hermitian Adjoint of differential operator
I came across this equation (identity) (Eq. 4 in this paper):
$\int(-i d\psi/dx)^*\psi dx = \int \psi^*(-i d\psi/dx) dx + id(\psi^*\psi)/dx\mid_{-\infty}^{+\infty}$
I have trouble proving it. I ...
2
votes
1answer
57 views
Quantum mechanical analogue of conjugate momentum
In classical mechanics, we define the concept of canonical momentum conjugate to a given generalised position coordinate. This quantity is the partial derivative of the Lagrangian of the system, with ...
0
votes
1answer
150 views
Definition of energy
What is the definition of energy $E$ given a dispersion relation $\omega=\omega(k)$ where $k=|\vec k|$ and $\omega$ is not necessarily linearly proportional to $k$? What about momentum $\vec p$?
This ...
1
vote
0answers
65 views
about wavefunction and vector entries
I am beginer of physics and I am studying some very fundamental idea of quantum mechanics by myself. In the introducing book I am reading, there is an example to show a particle diffraced by a slit or ...
3
votes
4answers
384 views
Is the momentum operator well-defined in the basis of standing waves?
Suppose I want to describe an arbitrary state of a quantum particle in a box of side $L$. The relevant eigenmodes are those of standing waves, namely
$$ \left<x|n\right>=\sqrt{\frac{2}{L}}\cdot ...
3
votes
2answers
154 views
Does the canonical commutation relation fix the form of the momentum operator?
For one dimensional quantum mechanics $$[\hat{x},\hat{p}]=i\hbar $$
Does this fix univocally the form of the $\hat{p}$ operator? My bet is no because $\hat{p}$ actually depends if we are on ...
3
votes
1answer
295 views
Momentum as Generator of Translations
I understand from some studies in mathematics, that the generator of translations is given by the operator $\frac{d}{dx}$.
Similarly, I know from quantum mechanics that the momentum operator is ...
3
votes
4answers
588 views
Intuitive explanation of why momentum is the Fourier transform variable of position?
Does anyone have a (semi-)intuitive explanation of why momentum is the Fourier transform variable of position?
(By semi-intuitive I mean, I already have intuition on Fourier transform between ...
0
votes
1answer
90 views
Would synchronized dancing be a good way to describe entangled atoms to a laymen?
I was talking my professor about entanglement swapping between light and matter and it is briefly described here:
You start out with a crystal capable of doing parametric down conversion of incoming ...
3
votes
2answers
161 views
Derivative of a Position Eigenket
I was flicking through Zettili's book on quantum mechanics and came across a 'derivation' of the momentum operator in the position representation on page 126. The author derived that ...
0
votes
1answer
293 views
Momentum in quantum mechanics
In quantum mechanics, we can have some superposition of matter waves that have different wavelengths. If then, can't momentum of a particle change every time measurement takes place? Or should I ...
4
votes
4answers
499 views
Uncertainty Principle for a Totally Localized Particle
If a particle is totally localized at $x=0$, its wave function $\Psi(x,t)$ should be a Dirac delta function $\delta(x)$. Accordingly, its Fourier transform $\Phi(p,t)$ would be a constant for all $p$, ...
0
votes
2answers
482 views
Matter waves and de Broglie wave length
The wavelength of a particle of momentum p is calculated using De Broglie relation.
The de Broglie relation was postulated for what is called a matter waves. Now according to the statistical ...
4
votes
2answers
419 views
Use of Operators in Quantum Mechanics
I understand the form of operators in use for quantum mechanics such as the momentum operator:
$$\hat{\text{P}}=-ih\frac{d}{dx}$$ My question is in what ways can I use it and what am I getting back? ...
6
votes
3answers
257 views
Lorentz force in Dirac theory and its classical limit
It is well known that in Dirac theory the time derivative of $P_i=p_i+A_i$ operator (where $p_i=∂/∂_i$, $A_i$ - EM field vector potential) is an analogue of the Lorentz force:
$\frac{dP_i}{dt} = ...
12
votes
4answers
573 views
Energy is actually the momentum in the direction of time?
By comparatively examining the operators
a student concludes that `Energy is actually the momentum in the direction of time.' Is this student right? Could he be wrong?
3
votes
2answers
848 views
Quantum momentum (De Broglie)
The de broglie hypothesis suggests a particle can be associated with a wave of
momentum $p = \hbar k$
my question is the following: how does one arrive at this concept of the momentum of a wave?
I ...
3
votes
1answer
663 views
Confusion between the de Broglie wavelength of a particle and wave packets
So I learned that the de Broglie wavelength of a particle, lambda = h/p, where h is Planck's constant and p is the momentum of the particle. I also learned that a quantum mechanics description of a ...
4
votes
1answer
205 views
Uncertainly Principle in orthogonal directions
The Heisenberg Principle states that for each direction, $\Delta x\cdot \Delta p_x \ge \hbar , \Delta y\cdot \Delta p_y \ge \hbar$ and $\Delta z\cdot \Delta p_z \ge \hbar$.
But, can anything be said ...
3
votes
2answers
908 views
Momentum-Representations in Quantum Mechanics
Why do we get information about position and momentum when we go to different representations. Why is momentum, which was related to time derivative of position in classical physics, now in QM just a ...
