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2. Let x(t) be the coordinate operator for a free particle in one dimension in the...
Problem 3: A free particle of mass m in one dimension is in the state Hbr...
Problem 3: A free particle of mass m in one dimension is in the state Hbr Ψ(z, t = 0) = Ae-ar with A, a and b real positive constants. a) Calculate A by normalizing v. b) Calculate the expectation values of position and momentum of the particle at t 0 c) Calculate the uncertainties ΔΧ and Δ1) for the position and momentum at t 0, Do they satisfy the Heisenberg relation? d) Find the wavefunction Ψ(x, t) at a...
A particle of mass m moves in one dimension. Let x(t) denote the position of the...
A particle of mass m moves in one dimension. Let x(t) denote the position of the particle at time t. The particle is subjected to a force which depends only on the position of the particle; when the particle is at position x, the force is -A sin(Bx), where A and B are some positive constants. Fill in the blank so that we end up with the differential equation that describes the motion: x" = Note that x = 0...
qm 09.2 2. (i) In one dimension, the momentum operator is given by d Ô =...
qm 09.2
2. (i) In one dimension, the momentum operator is given by d Ô = -ih- dx Determine the x dependence of the (un-normalised) momentum eigenfunction for a particle of momentum p, free to move along the x axis. [4 marks] (ii) A particle that is free to move along the x axis is described by a wavefunction v(x) = 1/ va, 0, |x<a/2 1x1 >a/2. (a) Show that the probability of measuring a momentum between p and p...
The initial wave function of a free particle is: Ψ(x,0) = A, for |x| = 0,...
The initial wave function of a free particle is: Ψ(x,0) = A, for |x| = 0, otherwise where a and A are positive real numbers. The particle is in a zero (or constant) potential environment since it is a free particle a) Determine A from normalization. b) Determine φ(p) = Φ(p,0), the time-zero momentum representation of the particle state. What is Φ(p,t)? Sketch φ(p). Locate the global maximum and the zeros of φ(p). Give the expression for the zeros (i.e.,...
Consider a particle confined to one dimension and positive z with the wave function 0 where...
Consider a particle confined to one dimension and positive z with the wave function 0 where N is a real normalization constant and α is a real positive constant with units of (length)-1. For the following, express your answers in terms of α: f) Calculate the expectation value of the momentum, (p) via the canonical expression -0o g) Calculate the expectation value of (p) via the canonical expression h) Use your results for(i) and (pay to calculate the variance in...
A particle undergoes simple harmonic motion (SHM) in one dimension. The r coordinate of the particle...
A particle undergoes simple harmonic motion (SHM) in one dimension. The r coordinate of the particle as a function of time is r(t)Aco() where A is the called the amptde" and w is called the "angular frequency." The motion is periodic with a period T given by Many physical systems are described by simple harmonic motion. Later in this course we will see, for example, that SHM describes the motion of a particle attached to an ideal spring. (a) What...
3. A particle of mass m moves in one dimension, and has position r(t) at time t. The particle has...
Mechanics.
3. A particle of mass m moves in one dimension, and has position r(t) at time t. The particle has potential energy V(x) and its relativistic Lagrangian is given by where mo is the rest mass of the particle and c is the speed of light (a) Writing qr and denoting by p its associated canonical momenta, show that the Hamiltonian is given by (show it from first principles rather than using the energy mzc2 6 marks (b) Write...
234 = 9 marks ] Question 3 Work in 1-dimension for simplicity. Let ф(2', t) %3DT(2",...
234 = 9 marks ] Question 3 Work in 1-dimension for simplicity. Let ф(2', t) %3DT(2", 2)ф(т, t), where T(',x) is called the "space translation operator from x to x". so that r' = x + dx. Explain why the following form makes (a) Consider an infinitesimal space translation, sense for the corresponding "infinitesimal space translation operator": idx T(xdr, x) 1 + ô+ (dx2) (b) Hence show that O is the momentum operator. (c) Explain why the full space translation...
Consider a particle with mass m described by the Hamilton operator for a one-dimensional harmonic oscillator...
Consider a particle with mass m described by the Hamilton operator for a one-dimensional harmonic oscillator 2 Zm 2 The normalized eigenfunctions for Hare φη (x) with energies E,,-(n + 2) ha. At time t-0 the wavefunction of the particle is given by у(x,0)- (V3іфі (x) + ф3(x)). Now let H' be an operator given by where k is a positive constant. 1) Show that H' is Hermitian. 2) Express H' by the step-operators a+ and a 3) Calculate the...
A free particle moving in one dimension has wave function Ψ(x,t)=A[ei(kx−ωt)−ei(2kx−4ωt)] where k and ω are...
A free particle moving in one dimension has wave function Ψ(x,t)=A[ei(kx−ωt)−ei(2kx−4ωt)] where k and ω are positive real constants. At t = π/(6ω) what are the two smallest positive values of x for which the probability function |Ψ(x,t)|2 is a maximum? Express your answer in terms of k.
Problem 3: A free particle of mass m in one dimension is in the state Hbr Ψ(z, t = 0) = Ae-ar with A, a and b real positive constants. a) Calculate A by normalizing v. b) Calculate the expectation values of position and momentum of the particle at t 0 c) Calculate the uncertainties ΔΧ and Δ1) for the position and momentum at t 0, Do they satisfy the Heisenberg relation? d) Find the wavefunction Ψ(x, t) at a...
A particle of mass m moves in one dimension. Let x(t) denote the position of the particle at time t. The particle is subjected to a force which depends only on the position of the particle; when the particle is at position x, the force is -A sin(Bx), where A and B are some positive constants. Fill in the blank so that we end up with the differential equation that describes the motion: x" = Note that x = 0...
qm 09.2
2. (i) In one dimension, the momentum operator is given by d Ô = -ih- dx Determine the x dependence of the (un-normalised) momentum eigenfunction for a particle of momentum p, free to move along the x axis. [4 marks] (ii) A particle that is free to move along the x axis is described by a wavefunction v(x) = 1/ va, 0, |x<a/2 1x1 >a/2. (a) Show that the probability of measuring a momentum between p and p...
Consider a particle confined to one dimension and positive z with the wave function 0 where N is a real normalization constant and α is a real positive constant with units of (length)-1. For the following, express your answers in terms of α: f) Calculate the expectation value of the momentum, (p) via the canonical expression -0o g) Calculate the expectation value of (p) via the canonical expression h) Use your results for(i) and (pay to calculate the variance in...
A particle undergoes simple harmonic motion (SHM) in one dimension. The r coordinate of the particle as a function of time is r(t)Aco() where A is the called the amptde" and w is called the "angular frequency." The motion is periodic with a period T given by Many physical systems are described by simple harmonic motion. Later in this course we will see, for example, that SHM describes the motion of a particle attached to an ideal spring. (a) What...
Mechanics.
3. A particle of mass m moves in one dimension, and has position r(t) at time t. The particle has potential energy V(x) and its relativistic Lagrangian is given by where mo is the rest mass of the particle and c is the speed of light (a) Writing qr and denoting by p its associated canonical momenta, show that the Hamiltonian is given by (show it from first principles rather than using the energy mzc2 6 marks (b) Write...
234 = 9 marks ] Question 3 Work in 1-dimension for simplicity. Let ф(2', t) %3DT(2", 2)ф(т, t), where T(',x) is called the "space translation operator from x to x". so that r' = x + dx. Explain why the following form makes (a) Consider an infinitesimal space translation, sense for the corresponding "infinitesimal space translation operator": idx T(xdr, x) 1 + ô+ (dx2) (b) Hence show that O is the momentum operator. (c) Explain why the full space translation...
Consider a particle with mass m described by the Hamilton operator for a one-dimensional harmonic oscillator 2 Zm 2 The normalized eigenfunctions for Hare φη (x) with energies E,,-(n + 2) ha. At time t-0 the wavefunction of the particle is given by у(x,0)- (V3іфі (x) + ф3(x)). Now let H' be an operator given by where k is a positive constant. 1) Show that H' is Hermitian. 2) Express H' by the step-operators a+ and a 3) Calculate the...
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