Free particle Definition and 145 Threads

I have a question about an example about the choice of the operators needed to describe a system, the text is reported below: "3D systems with ##H = (p_1^2+ p_2^2+ p_3^2)/(2m)## but no potential. Classically, the number of degrees of freedom is 6 corresponding to the six canonical variables xi...

Hi, reading "Mechanics" book by Landau-Lifshitz, they derive from spatial homogeneity that the Lagrangian ##L## of a free particle cannot explicitly depend on spatial coordinates ##q## in an inertial frame. However my point is as follows: suppose to consider the Lagrangian ##L= \frac 1 2...

I had an interesting thought. Let's only look at the free particle scenario. We derive euler lagrange even without the need to know what exactly ##L## is (whether its a function of kinetic energy or not) - deriving EL still can be done. Though, because in the end, we end up with such...

Abstract: If a laser shoots photons at a pinhole with a screen behind it, we get a circular non-interference pattern on the screen. Is this distribution Guassian, and if not, what would its wave function be? ===================== Assume a double-slit like experiment, but instead of double...

Mine is a simple question, so I shall keep development at a minimum. If a particle is moving in the absence of a potential (##V(x) = 0##), then ##\frac{\langle\hat p \rangle}{dt} = \langle -\frac{\partial V}{\partial x}\rangle=0## will require that the momentum expectation value remains...

Summary: The initial problem states: Consider a free particle of mass m moving in one space dimension with velocity v0. Its starting point is at x = x0 = 0 at time t = t0 = 0 and its end point is at x = x1 = v0t1 at time t = t1 > 0. and this info is to do the 3 problems written out. a)...

Hartle, Gravity "An observer in an inertial frame can discover a parameter ##t##with respect to which the positions of all free particles are changing at constant rates. This is time" Then goes on to say "Indeed, inertial frames could be defined as Cartesian reference frames for which Newton’s...

The textbook I am self studying says that the wave function for a free particle with a known momentum, on the x axis, can be given as Asin(kx) and that the particle has an equal probability of being at any point along the x axis. I understand the square of the wave function to be the probability...

So I think I use the right approach and I get uncertainty like this: And it's interval irrelevant(ofc), So what kind of wave function gives us \h_bar / 2 ? I guess a normal curve? if so, why is normal curve could be? if not then what's kind of wave function can reach the lower bound

If ##\psi(x, t)=\left(\frac{1}{2 \pi \alpha^{2}}\right)^{1 / 4} \frac{1}{\sqrt{\gamma}} e^{i p_{0}\left(x-p_{0} t / 2 m\right) / \hbar} e^{-\left(x-p_{0} t / m\right)^{2} / 4 \alpha^{2} \gamma}##where * ##\gamma=1+\frac{i t} {\tau}##( a complex number) * ##\tau=\frac{m h}{2...

For the free particle in QM, the energy and momentum eigenstates are not physically realizable since they are not square integrable. So in that sense a particle cannot have a definite energy or momentum. What happens during measurement of say momentum or energy ? So we measure some...

To me, the ##K## obtained by solving the Schrodinger equation and the de broglie wavelength seem two completely unrelated quantities. Can someone explain why have we equated ##K## and ##\frac{2\pi}{\lambda}##. Also, isn't writing ##p = \hbar K## implying that eigenstate of energy is also an...

Here we are talking about non-relativistic quantum physics. So we all know kinetic energy T = E - V = \frac{1}{2}mv^2 in classical physics. Here V is the potential energy of the particle and E is the total energy. Now what I am seeing is that this exact same relation is being used in quantum...

I think it is quite simple as an exercise, following the two relevant equations, but at the beginning I find myself stuck in going to identify the lagrangian for a relativistic system of non-interacting particles. For a free relativistic particle I know that lagrangian is...

Consider a free particle with rest mass ##m## moving along a geodesic in some curved spacetime with metric ##g_{\mu\nu}##: $$S=-m\int d\tau=-m\int\Big(\frac{d\tau}{d\lambda}\Big)d\lambda=\int L\ d\lambda$$...

Hi! I'm studying Shankar's Principle of quantum mechanics I didn't get the last conclusion, can someone help me understand it, please. Where did the l over rho come from?

EQ 1: Ψ(x,0)= Ae-x2/a2 A. Find Ψ(x,0) So I normalized Ψ(x,0) by squaring the function, set it equal to 1 and getting an A I. A=(2/π)¼ (1/√a) B. To find Ψ(x,t) EQ:2 Ψ(x,t)= 1/(√2π) ∫ ∅(k) ei(kx-ωt)dk --------->when ω=(ħk2)/2m and integral from -∞ to +∞ EQ 3: ∅(k)= 1/(√2π) ∫ Ψ(x,0)...

In quantum mechanics, the Eigenfunction resulting from the Hamiltonian of a free particle in 1D system is $$ \phi = \frac{e^{ikx} }{\sqrt{2\pi} } $$ We know that a function $$ f(x) $$ belongs to Hilbert space if it satisfies $$ \int_{-\infty}^{+\infty} |f(x)|^2 dx < \infty $$ But since the...

I've got the solution to the question but I just need more detail. I can't work out the first step of the solution to the second step. That should read, I don't know what they multiplied ih-bar by to make it (i/h-bar)^2?

It's been a long time since my last exam on QM, so now I'm struggling with some basic concept that clearly I didn't understand very well. 1) The Sch. Eq for a free particle is ##-\frac {\hbar}{2m} \frac {\partial ^2 \psi}{\partial x^2} = E \psi## and the solutions are plane waves of the form...

In Landau mechanics it's been given that multiple Lagrangians can be defined for a system which differ by a total derivative of a function. This statement is further used for the following discussion. I understand that the term for L has been expanded as a Taylor series but I can't understand...

Homework Statement From Griffiths GM 3rd p.266 Consider a free particle of mass ##m##. Show that the position and momentum operators in the Heisenberg picture are given by$$ {\hat x}_H \left( t \right) ={\hat x}_H \left( 0 \right) + \frac { {\hat p}_H \left( 0 \right) t} m $$ $$ {\hat p}_H...

Hi. I have just looked at a question concerning a free particle on a circle with ψ(0) = ψ(L). The question asks to find a self-adjoint operator that commutes with H but not p. Because H commutes with p , i assumed there was no such operator. The answer given , was the parity operator. It acts...

The action for a relativistic point particle is baffling simple, yet I don't really understand why it is written as, $$S = -m\int ds $$ I know it's right because we get the right equations of motion from it, but can one understand it in a more intuitive way?

Why for the free particle, the group velocity and phase velocity are not the same while we have only one wave? What is the envelope here?

I have a problem finding ##\left|Ψ(x,t)\right|^2## from the following equation: $$Ψ(x,t) = \frac 1 {\pi \sqrt{2a}} \int_{-∞}^{+∞} \frac {\sin(ka)} k e^{i(kx - \frac {ħk^2} {2m} t)} dk$$ and tried to plot like the pic below (Source Introduction to quantum mechanics by David. J. Griffiths, 2nd...

Hello physics forums. I'm trying to solve an old exam question. Would love your help. Homework Statement A free particle in one dimension is described by: ## H = \frac{p^2}{2m} = \frac{\hbar}{2m}\frac{\partial^2}{\partial x^2}## at ##t = 0## The wavefunction is described by: ## \Psi(x,0) =...

Suppose we have a particle of mass ##m## moving freely in the xy-plane, except for being constrained by hard walls to have ##-L/2 < y < L/2##. Now, the energy eigenstates would be something like ##\psi (x,y) = C \psi_x (x) \psi_y (y) = C e^{-ikx}\cos\left(\frac{n\pi y}{L}\right) ##, where...

So there's a free particle with mass m. \begin{equation} \psi(x,0) = e^{ip_ox/\hbar}\cdot\begin{cases} x^2 & 0 \leq x < 1,\\ -x^2 + 4x -2 & 1 \leq x < 3,\\ x^2 -8x +16 & 3 \leq x \leq 4, \\ 0 & \text{otherwise}. \end{cases} \end{equation} What does each part of the piecewise represent...

Homework Statement I am given the Hamiltonian of the relativistic free particle. H(q,p)=sqrt(p^2c^2+m^2c^4) Assume c=1 1: Find Ham-1 and Ham-2 for m=0 2: Show L(q,q(dot))=-msqrt(1-(q(dot))^2/c^2) 3: Consider m=0, what does it mean? Homework Equations Ham-1: q(dot)=dH/dp Ham-2: p(dot)=-dH/dq...

Homework Statement Hi, i have this problem: In a 3D space, a free particle is described by : $$ \Psi = Ne^{-ar} $$ with $$ r=| \vec r | $$ at the time t=0 . How can we write the wave function whit $$ \hbar \vec k $$ ?Homework EquationsThe Attempt at a Solution I know how to resolve this...

Homework Statement [/B] 1) I don't quite understand what 2.94 means on its own. It was derived from 2.93, yet it doesn't show a superposition of any sort. The author then takes 2.94, and attempts to normalise it by stating ##\int \Psi_k^* \Psi_k dx = \mid A^2 \mid\int dx = \infty ## What...

Given a source of electrons, like from an electron gun. Physicists call these freely traveling particles and often use a Gaussian wave packet to represent them with the group velocity being precisely defined as the velocity of the center of the packets. But if we do not measure the position of...

Homework Statement I want to plot a graph of the survival probability of the initial state ψ = e-|x| for a free particle. Hopefully this will enable me to plot some more difficult examples like the inverted oscillator etc for a project but I'm struggling fundamentally with the free particle...

Homework Statement The eigenstates of the momentum operator with eigenvalue k are denoted by |k>, and the state of the system at t = 0 is given by the vector |{ψ}>=\int \frac {dk}{2π} g(k)|{k}> Find the state of the system at t, |ψ(t)>. Compute the expectation value of \hat{P}. Homework...

Homework Statement The wave function is given as Ψ(x,t) = Ae^[i(k1x-ω1t)] + Ae^[i(k2x-ω2t)]. Show that particle average velocity Vav = ħ(k1+k2)/2m equals ω2-ω1/k2-k1. Average momentum of the particle is Pav = ħ(k2+k1)/2. Homework Equations p = ħk E=ħω K = 1/2 * mV^2 The Attempt at a Solution...

Hi everybody, I was reading about the free particle in a textbook and I got confused by the line: "If we adopt the convention that k and k are real, then the only oscillating exponentials are the eigenfuntions with positive energy" [Also see the attached picture with the...

Hi! I'm currently studying Griffith's fantastic book on QM, and I'm confused for a bit about the wave function for a free particle. Here's what I think so far; for a free particle, there are no stationary states, so therefore we can't solve the SE with...

Hello, When we normalize the free particle by putting it in a box with periodic boundary conditions, we avoid the "pathological" nature of the momentum representation that take place in the normal problem of a particle in a box with the usual boundary conditions of Ψ=0 at the two borders. Thus...

It would be really appreciated if somebody could clarify something for me: I know that stationary states are states of definite energy. But are all states of definite energy also stationary state? This question occurred to me when I considered the free particle(plane wave, not a Gaussian...

Hello! Could somebody please tell me how i can compute the expectation value of the momentum in the case of a free particle(monochromatic wave)? When i take the integral, i get infinity, but i have seen somewhere that we know how much the particle's velocity is, so i thought that we can get it...

Hello! When we are dealing with a free particle in spherical coordinates,the position eigenfunction of the free particle is \psi_{klm}(r,\phi,φ)=\langle r\phiφ|klm\rangle=J_{l}(kr)Y_{lm}(\phi,φ). Here appears that the wavefunction describe a free particle of energy Ek of well-defined angular...

Homework Statement 5) A free particle moving in one dimension is in the state Ψ(x) = ∫ isin(ak)e(−(ak)2/2)e(ikx) dk a) What values of momentum will not be found? b) If the momentum of the particle in this state is measured, in which momentum state is the particle most likely to be found? c)...

Homework Statement By finding the Lagrangian and using the metric: \left(\begin{array}{cc}R^2&0\\0&R^2sin^2(\theta)\end{array}\right) show that: \theta (t)=arccos(\sqrt{1-\frac{A^2}{\omega^2}}cos(\omega t +\theta_o)) Homework EquationsThe Attempt at a Solution So I got the lagrangian to be...

When dealing with Dirichlet boundary conditions, that is asking for the wavefunction to be exactly zero at the boundaries, it can be clearly seen that (0,0,0) is not a physical situation as it is not normalizable. (Wavefunction becomes just 0 then) However when dealing with periodic boundary...

The free particle (zero potential) solution of the Schrodinger equation for a plane wave (from my understanding) is ψ= A e^(ikx) + B e^(-ikx). I have 2 questions: 1) This is the so-called free "particle" solution for a plane wave. Does this same solution apply to a light wave/photon also...

Shankar ("Principles of Quantum Mechanics", 2nd ed.) shows that the free particle propagator "matrix element" is given by (see p. 153): ## \qquad \langle x | U(t) | x' \rangle = U(x,t;x') = \left(\frac{m}{2\pi\hbar it}\right)^{1/2} e^{im(x-x')^2/2m\hbar} ##, which can be used to evaluate the...

Can someone help with what must be a simple math issue that I'm stuck on. Shankar ("Principles of Quantum Mechanics" p. 153) evaluates the propagator for a free particle in Equation 5.1.10. A scan of the chapter is available here...

I am hung up on what must be a very elementary matter, but I’m unable to see where I’m wrong. I reference R. Shankar's "Principles of Quantum Mechanics". For the free particle with https://www.physicsforums.com/file:///C:/Users/DANIEL~1.ABR/AppData/Local/Temp/msohtmlclip1/01/clip_image002.png ...

The wavefunction ##\Psi(x,t)## for a free particle is given by ##\Psi(x,t) = A e^{i(kx-\frac{\hbar k}{2m}t)}## This wavefunction is non-normalisable. Does this mean that free particles do not exist in nature?