0
votes
2answers
88 views

Determine $p_x$ from $[x,p_x]=i\hbar $ [closed]

With $[x,p_x]=i\hbar $, how to determine the form of the operator $p_x$?
3
votes
1answer
34 views

Probability of measuring momentum [closed]

Suppose we have this wavefunction: $$ \psi = A \left( cos(kx) + cos (2kx) \right) $$ I have to find the possible results of measurement of momentum and their probabilities. Attempt For a momentum ...
1
vote
1answer
71 views

Inner product of position and momentum eigenkets

Let's define $\hat{q},\ \hat{p}$ the positon and momentum quantum operators, $\hat{a}$ the annihilation operator and $\hat{a}_1,\ \hat{a}_2$ with its real and imaginary part, such that $$ \hat{a} = ...
0
votes
2answers
35 views

Difference between momentum calculated from wavefunction and momentum computed by energy formula

Take a quantum well, for example. The energy of an electron is given by: $E_n=k_n n^2$. Moreover, we know that the momentum of an electron with energy $E_n$ is given by: $p=mv=m(2T/m)^{1/2}$, where ...
2
votes
1answer
76 views

Does the average momentum vanish for an eigenstate of the simple harmonic oscillator?

Suppose we have a simple harmonic oscillator, let's consider the ground state, $|0\rangle$ and the first excited state $|1\rangle$. $\langle 0|\hat p|0 \rangle$ is zero right? Since the particle can ...
1
vote
0answers
69 views

Particle In a Box and momentum, velocity

So on a homework assignment, we are give the width of a well, $a$, and the mass of the particle $m$ and we want to find the average velocity of the particle at the n=1 state. So here is my attempt at ...
0
votes
1answer
75 views

Quantum Mechanics - Finding momentum probability density [closed]

everyone. I got a bit stuck on 2(iii), this is supposed to be a easy question, but i don't know how you get the square term? I thought you just do the Fourier transform, but then I got some ...
-1
votes
1answer
84 views

How does $p_x$ commute with $p_y$, i.e. $[p_x,p_y]=0$? [closed]

I know it's a simple and basic question but would someone show me how to evaluate $[\hat{p}_x,\hat{p}_y]$?
3
votes
3answers
182 views

Why is momentum quantized in a 1D box even though the operator doesn't give eigenstates?

We don't get eigenstates of momentum when we operate momentum operator in the wave function of particle in a 1D box problem yet we say momentum is quantized in this situation. Why is it so?
4
votes
4answers
248 views

Complex conjugate of momentum operator

Consider momentum operator representation in position space. $$\hat{p}=-i\frac{\partial}{\partial x} \,\ \text{and its eigen functions are } e^{ipx} \,\text{and} \,\ e^{-ipx}.$$ ...
0
votes
2answers
64 views

Measurement of Mass and Momentum of a particle simultaneously

In quantum mechanics can the mass and the linear momentum of a particle be measured precisely or do they commute ?
1
vote
3answers
132 views

What happens with a tunneling particle when its momentum is imaginary in QM?

In classical mechanics the motion of a particle is bounded if it is trapped in a potential well. In quantum mechanics this is no longer the case and there is a non zero probability of the particle to ...
2
votes
1answer
87 views

In the Dirac equation, do $\alpha$ and $p$ commute?

The Dirac Hamiltonian is given as $H = \vec \alpha·\vec pc + \beta mc^2$ , Do the alpha and beta operators commute with the momentum operator? If yes then how?
6
votes
2answers
193 views

Why $-i\hbar\vec\nabla$ for momentum in quantum mechanics, while $m\vec{v}$ in classical mechanics?

I am a little bit confused when thinking of the momentum representation in QM and CM. In QM, momentum is represented as $-i\hbar\vec\nabla$, while in classical, momentum is represented as $m\vec{v}$. ...
0
votes
2answers
79 views

Momentum Operator in Quantum Mechanics

1) What is the difference between these two momentum operators: $\frac{\hbar}{i}\frac{\partial}{\partial x}$ and $-i\hbar\frac{\partial}{\partial x}$? How are these two operators the same? My ...
0
votes
0answers
49 views

Including special relativistic effects in momentum in Heisenberg's Uncertainty Principle

I've been told that an electron is somewhere within the space of $10^{-10}m$ and am supposed to find the uncertainty in its velocity. Simply applying $m\Delta x \Delta v \geq \frac{h}{4\pi}$ results ...
0
votes
2answers
140 views

What is the reason behind why a quantum particle cannot be at rest?

So I've seen different reasonings for this; which is correct, or are they both corollaries of each other? 1) For a particle to be at rest, we would know its momentum and therefore by Heisenberg's ...
1
vote
2answers
110 views

Shortcut to find $\hat{p}^2$ expectation value

I have been going through several calculations where I am asked to calculate $\langle p^2 \rangle$ and the task is proving to be pretty tedious. Does anyone know of a shortcut for this? Such as with ...
0
votes
1answer
95 views

Quantum explanation of Newton's Third Law of Motion

Newton's law states that for every action there is equal and opposite reaction. This law explains how rockets fly in space and accounts for the the majority of the lift action generated by a ...
0
votes
0answers
87 views

Change of QM Momentum operator under coordinate transformation

Can any one please let me know what is the general procedure to construct the momentum operator under some coordinate transformation? For example, I understand that if ...
6
votes
1answer
89 views

In calculations with uncertainty principle why could you equate the uncertainty in momentum with the actual momentum of the system

This website is trying to calculate the confinement energy of a electron starting from the uncertainty principle, but it does this: $\Delta p=p$. Why is this valid?
8
votes
5answers
528 views

How to get the position operator in the momentum representation from knowing the momentum operator in the position representation?

I know that $$\tag{1}\hat{p}~=~-i\hbar \frac{\partial}{\partial x}~.$$ How can I get $$\tag{2}\hat{x}~=~i\hbar \frac{\partial}{\partial p}~?$$ I think this simple and I'm just over thinking it, ...
0
votes
1answer
319 views

How to derive or justify the expressions of momentum operator and energy operator?

It has been noted here$\! { \, }^{\text(1, 2)}$, for instance, that $$\mathbf{F} = \frac{d}{dt}\!\!\biggl[ \, \mathbf{p} \, \biggr]$$ is true in all contexts. Likewise, in notable contexts it is ...
0
votes
1answer
288 views

Harmonic Oscillator Energy to Momentum Expectation Value

If we are given a wave function written in terms of harmonic oscillator energy eigenfunctions how can we determine the maximum possible momentum expectation value? It's a combination of the first two ...
3
votes
2answers
87 views

Identity in quantum operator tutorial

I'm reading this tutorial by Ben Simons entitled Operator methods in quantum mechanics in connection with his course in advanced QM, and I'm a bit puzzled by an identity in page 25, a bit above ...
1
vote
1answer
3k views

Proof that the momentum operator is Hermitian

I am trying to prove that the momentum $p_x$ operator is Hermitian, my approach is the following $$<p_x>~=~\int \Psi^*(\vec{r},t)[-ih\frac{\partial}{\partial x}]\Psi(\vec{r},t)\, d^3r.$$ I try ...
0
votes
2answers
1k views

What is the Momentum Operator?

I know the equation for the momentum operator, but what exactly is the momentum operator? It's bizarre to me that taking the derivative of the wave function, which is an operator, should return ...
4
votes
3answers
429 views

How does the momentum operator act on state kets?

I have been going through some problems in Sakurai's Modern QM and at one point have to calculate $\langle \alpha|\hat{p}|\alpha\rangle$ where all we know about the state $|\alpha\rangle$ is that ...
1
vote
3answers
431 views

The Momentum Operator in QM

I've seen the 'derivation' as to why momentum is an operator, but I still don't buy it. Momentum has always been just a product $m{\bf v}$. Why should it now be an operator. Why can't we just multiply ...
4
votes
2answers
280 views

What is the most general expression for the coordinate representation of momentum operator?

I have a question about deriving the coordinate representation of momentum operator from the commutation relation, $[x,p]= i$. One derivation (ref W. Greiner's Quantum Mechanics: An Introduction, 4th ...
-2
votes
2answers
201 views

Determine whether the ground state is an eigenfunction of [p] and of [p^2] [closed]

Consider a particle confined in an infinite square well potential of width L, $$V(x)=\left\{ \begin{array}{ll}\infty, &{\rm for}\ (x \le 0)\vee (x \ge L) \\0, &{\rm for} \ 0 < x < L ...
-3
votes
2answers
296 views

Momentum of a particle? [closed]

I really need help to understand what is momentum of a particle (of a photon, proton, an electron...) I see so many definitions! My main questions are: •What exactly is momentum •What are the ...
0
votes
1answer
791 views

Expectation value of momentum

I'm having a problem with an expectation value that doesn't seem to add up for me. What I know is, that $\psi(\vec{r})$ is a wavefunction for a particle in three dimensions. The Hamiltonian is given ...
5
votes
3answers
298 views

How can particles travel in a straight line?

A particle can be set off in a certain direction by giving them momentum. Momentum is a vector, so the particle heads off in a specific direction. But the wave function of the particle allows it to ...
1
vote
1answer
638 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
380 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 ...
1
vote
0answers
76 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
773 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 ...
4
votes
3answers
394 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 ...
4
votes
1answer
2k 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 ...
4
votes
4answers
2k 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
155 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
347 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
444 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
754 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 $p$, ...
2
votes
2answers
1k 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
671 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
510 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
888 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
2k 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 ...