The dynamics of a particle moving one-dimensionally in a potential V (x) is governed by the Hamiltonian Ho = p²/2m + V (x), where p = is the momentuin operator. Let E, n = of Ho. Now consider a new Hamiltonian H given parameter. Given A, m and E, find the eigenvalues of H. -ih d/dx 1, 2, 3, ... , be the eigenvalues Ho + Ap/m, where A is a %3|
Q: What is the Hamiltonian of a system? Show that the kinetic energy operator commutes with the linear…
A: The momentum operator in 1D is defined as px=-ihδδx the Kinetic energy operator is Tx=-h22mδ2δx2…
Q: 3. The Hamiltonian has discrete nondegenerate eigenvalues Eñ, n = 1, 2, . . .. What is the general…
A: The objective of the question is to find the general solution of the time-dependent Schrödinger…
Q: Two masses and 3 springs. нотот Consider the longitudinal oscillations, i.e., along the axis, of a…
A:
Q: A point particle moves in space under the influence of a force derivable from a generalized…
A: The generalized force equation is Qj = −∂U/∂qj + d/dt (∂U/∂q˙j).
Q: . A particle of mass m in 3D is subject to a potential V(x, y, z) = ²x Write the Lagrangian and the…
A:
Q: 2 ô Consider operator Ô 1 and function R(r)= e br. What must be the r ôr value of constant b for…
A: The operator is given as, O^=-∂2∂r2+2r∂∂r-1r The function is given as, R(r)=e-br Applying the…
Q: A system, initially in state li), is disturbed H' (t) = G sin wt with the time-independent G…
A: According to given data, system initially in |i> state experience disturbance H'=G sinωt The…
Q: The Hamiltonian of a system has the form 1 d² 2 dx² अ = • 1⁄2 x² + √4x¹ = Ĥ0 + V4Xª Let ½(x) = |n)…
A:
Q: Problem #1 (Problem 5.3 in book). Come up with a function for A (the Helmholtz free energy) and…
A:
Q: A Hamiltonian is given in matrix form as 0 Ĥ。 (19) ħwo (1 2 0 a. What are the energy eigenvalues? b.…
A:
Q: Show that the function y = cos(4x) Is an eigen function for this operator d? Â = dx2
A: Given, wave function ψ=cos4x
Q: Consider a spin-1 particle with Hamiltonian Ĥ = AS² + B(Ŝ² − S²). Assume B < A, treat the second…
A: The unperturbed Hamiltonian for a spin-1 particle is: H_0 = AS_Z^2 where S_Z is the z-component of…
Q: Sketch the z-plane pole-zero plot and determine the stability status for the following digital…
A:
Q: Let VA), B) be the eigenvectors of the Hamiltonian ♬ of a two-level system Ĥ|VA,B) = EA,B|VA,B) EA>…
A:
Q: Consider the Hamiltonian Ĥ = ¸+ Ĥ' where E 0 0 Ĥ₁ 0 E 0 and Ĥ' is the time independent perturbation…
A:
Q: Consider the basis S = {v1, V2} for R2, where v1 = (1, 1) and v2 = (1, 0), and let T:R? – R2 be the…
A: Given: S=(v1,v2)v1=(1,1)v2=(1,0)
Q: Consider the "rigid rotor" by a m FA m 2 rigid rod, free to rotate in 3D. 2 masses connected
A:
Q: H.W. A one dimensional harmonic oscillator is described by the Hamiltonian Ĥ = ho â'௠+ -/-) as…
A:
Q: Consider a particle in an infinite potential well. (+∞o x a a. Write down the form of the…
A:
Q: Two mass points of mass m1 and m2 are connected by a string passing through a hole in a smooth table…
A:
Q: Suppose I have an operator Â, and I discover that Â(2²) = 5 sina and Â(sin x) = 5x². (a) Find Â(2²…
A: A^(x2)=5 sin xA^(sin x)=5 x2
Q: Assume MKS units... Let Q be an open subset of R³. Let B: Q -R³ be a continuous vector .field,…
A:
Q: A point particle moves in space under the influence of a force derivable from a generalized…
A: Classical Mechanics
Q: What is the value of the commutator [Sy , ž]? Here Jy is the y-component of the angular momentum…
A: using different properties of commutator we can solve the question
Q: (d) Consider the arbitrary ket |u)=i-1 uli), where i) is an orthonormal basis. i. Show that u =…
A:
Q: The Hamiltonian operator Ĥ for the harmonic oscillator is given by Ĥ = h d? + uw? â2, where u is the…
A:
Q: Show explicitly how to construct the L^3 operator. Then determine if the spherical harmonics (Yl,m)…
A:
Trending now
This is a popular solution!
Step by step
Solved in 3 steps with 3 images
- A prticle in the infinite square wen has its inidal wave fumtionan even mixture of the first two stationary states: Normalize IV(x. 0).Find IV(x, t) and IV(x,0)I2 Compute<x>. Notice that it oscillates in time. What is the angular frequencyof the oscillation? What is the amplitude of the oscillation? Compute <p>.If you measured the energy what values might you gets andwhat is the probability the valueOf H .HOW does it compare with E1and E2 *** Please pay attention to the picture***Two mass points of mass m1 and m2 are connected by a string passing through a hole in a smooth table so that m1 rests on the table surface and m2 hangs suspended. Assuming m2 moves only in a vertical line, what are the generalized coordinates for the system? Write the Lagrange equations for for the system and, if possible, discuss the physical significance any of them might have. Reduce the problem to a single second-order differential equation and obtain a first integral of the equation. What is its physical significance? (Consider the motion only until m1 reaches the hole.)(a) Derive the following general relation for the first order correction to the energy, E, in time-independent perturbation theory where the 0 are the eigen-functions of the unperturbed Hamiltonian Ho and H' is the time independent perturbation. (b) A particle moves in a potential given by Vo sin(ax/a) for 0 < x < a V : %D otherwise where Vo is a small constant. Treat this as a perturbation for the case of a particle in an infinitely deep square well of width a and calculate the change in energy of the lowest energy state to first order in Vo.
- Let Ymdenote the eigenfunctions of a Hamiltonian for a spherically symmetric potential V(r). The wavefunction + √104 213 ] is an eigenfunction of 21-1 (b) H and L₂ (c) H and L² = Y. +√√54. 210 4 (a) H₂ L² and L₂ (d) L² and L₂I solved it but I need help in two parts.For first part, How to show thev formula is a solution of ODE? Second, for the third part, how to show it is bounded because I can not integratw matrix?Write down the Hamiltonian function and Hamilton’s canonical equations for a simple Atwood machine.
- The Hamiltonian of an electron of mass m in a constant electric field E in one dimension can be written as Ĥ=+eEx where â and are the position and momentum operators, respectively. With initials conditions (t = 0) = 0 and p(t = 0) = 0, which one of the following gives (t) at time in the Heisenberg picture? You may use the commutator [â,p] = iħ. O a. O b. eEt2 2m O C. e Et O d. -eEt O e. eEt² m pt mThe eigenfunctions of Hermitian Operators are orthonormal (orthogonal and normalizable). a. Prove the eigenfunctions of the Hamiltonian Operator for a particle in a box that extends from x = 0 to x = a: плх y.(x) = are orthonormal by integrating a pair of functions, y,(x) and y.(x), with n = m in one case and n m in another. b. For the ground state of a particle in a box, use the momentum and position operators to show that the expectation values are 0 for momentum and - for position, by evaluating the resulting integrals. c. Calculate the uncertainty in x and p (0x and Op) for the lowest energy state of a particle in a box by taking the standard deviation of x and p as a measure of their uncertainty: Ar =0, = (x*)-(x)* and Ap=o,=Kp*)-(p)* Is the product you obtain for OxOp consistent with the Heisenberg Uncertainty Principle?please solve