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3. (Cohen-Tannoudji, page 635) This exercise is a continuation of the last problem. The two particle...
3. A particle is in a 1D box (infinite potential well) of dimension, a, situated symmetrically ab...
3. A particle is in a 1D box (infinite potential well) of dimension, a, situated symmetrically about the origin of the x-axis. A measurement of energy is made and the particle is found to have the ground state energy: 2ma The walls of the box are expanded instantaneously, doubling the well width symmetrically about the origin, leaving the particle in the same state. a) Sketch the initial potential well making it symmetric about x - 0 (note this is different...
2.2 Two-level system A particle in the box is described by the following wavefunction 1 1...
2.2 Two-level system A particle in the box is described by the following wavefunction 1 1 V(x, t) + V2 V2 = Um(x)e -i(Em/h) In other words, this state is a superposition of two modes: n-th, and m-th. A superposition that involves only two modes (not necessarily particle in the box modes, but any two modes) is called a "two-level system”. A more modern name for such a superposition is a "qubit”. a) Come up with an expression for the...
322 CHAPTER 5. ANGULAR MOMENTUM Problem 5.12 Consider a particle whose wave function is 1 222-x2-y2...
322 CHAPTER 5. ANGULAR MOMENTUM Problem 5.12 Consider a particle whose wave function is 1 222-x2-y2 4 A 3 xz (x, y, z) = 2 2 Calculate L2 (x, y, z) and L-y(x, y, z). Find the total angular momentum of this particle. (b) Calculate L+ y (x, y, z) and (Y L+ W). (c)If a measurement of the z-component of the orbital angular momentum is carried out, find the probabilities corresponding to finding the results 0, h, and -h....
please help 1. The eigenfunctions of a particle in a square two-dimensional box with side lengths...
please help
1. The eigenfunctions of a particle in a square two-dimensional box with side lengths a = b = L are non, (x, y) = { sin ("T") sin (9,7%) = xn, (x)}n, (y) where n. (c) and on, (y) are one-dimensional particle-in-a-box wave functions in the x and y directions. a. Suppose we prepare the particle in such a way that it has a wave function V (2,y) given by 26,0) = Võru (s. 1) + Vedra ....
3.9. A particle of mass m is confined in the potential well 0 0<x < L...
3.9. A particle of mass m is confined in the potential well 0 0<x < L oo elsewhere (a) At time t 0, the wave function for the particle is the one given in Problem 3.3. Calculate the probability that a measurement of the energy yields the value En, one of the allowed energies for a particle in the box. What are the numerical values for the probabilities of obtaining the ground-state energy E1 and the first-excited-state energy E2? Note:...
1. The wavefunction corresponding to Im> energy and angular momentum eigenstate of a particle rotating in...
1. The wavefunction corresponding to Im> energy and angular momentum eigenstate of a particle rotating in a ring for m-l and m--1 are, respectively N2T where ? is the angular position of the particle relative to thex axis (see slide 15 of lecture 74a). (a) show that the probability density does not depend on 0. (b) Show that P,(o)-sin() where p, (0) rticle in the quantum state V, (d) p, (0) obviously resembles one of the orbitals of the is...
Problem 3 (Needed for Problem 4) A continuous random variable X is said to have an...
Problem 3 (Needed for Problem 4) A continuous random variable X is said to have an exponential distribution, written Exp(X), if its probability density function f is such that le- if > 0 10 if x < 0 f(0) = 0 where > 0 is a real number. 1. Compute the mean of X 2. Compute the variance of X 3. Compute the cumulative distribution function F of X. Use this to show that for any real numbers s and...
Problem 8.3 - A New Two-State System Consider a new two-level system with a Hamiltonian given...
Problem 8.3 - A New Two-State System Consider a new two-level system with a Hamiltonian given by i = Ti 1461 – 12) (2) (3) Also consider an observable represented by the operator Ŝ = * 11/21 - *12/11: It should (hopefully) be clear that 1) and 2) are eigenkets of the Hamiltonian. Let $1) be an eigenket of S corresponding to the smaller eigenvalue of S and let S2) be an eigenket of S corresponding to the larger eigenvalue....
Suppose that the position of one particle at time t is given by Xi= 3 sint...
Suppose that the position of one particle at time t is given by Xi= 3 sint Yi = 2 cost 0 < t < 211 and the position of a second particle is given by X2 = -3+ cost y2 = 1+ sin t 0<t<21 (a)Graph the paths of both particles. How many points of ntersection are there? (b) Are any of these points of intersection collision points? In other words, are the particles ever at the same place at...
Use Python to solve each problem. 2. A particle moves according to a law of motion...
Use Python to solve each problem. 2. A particle moves according to a law of motion s = t 3 − 12t 2 + 24t, t ≥ 0. a) Find the velocity at time t. b) What is the velocity after 1 second? c) When is the particle at rest? d) Sketch the position function on t ∈ [0, 6] to determine when the particle is moving in the positive direction on that interval. e) Find the total distance traveled...
3. A particle is in a 1D box (infinite potential well) of dimension, a, situated symmetrically about the origin of the x-axis. A measurement of energy is made and the particle is found to have the ground state energy: 2ma The walls of the box are expanded instantaneously, doubling the well width symmetrically about the origin, leaving the particle in the same state. a) Sketch the initial potential well making it symmetric about x - 0 (note this is different...
2.2 Two-level system A particle in the box is described by the following wavefunction 1 1 V(x, t) + V2 V2 = Um(x)e -i(Em/h) In other words, this state is a superposition of two modes: n-th, and m-th. A superposition that involves only two modes (not necessarily particle in the box modes, but any two modes) is called a "two-level system”. A more modern name for such a superposition is a "qubit”. a) Come up with an expression for the...
322 CHAPTER 5. ANGULAR MOMENTUM Problem 5.12 Consider a particle whose wave function is 1 222-x2-y2 4 A 3 xz (x, y, z) = 2 2 Calculate L2 (x, y, z) and L-y(x, y, z). Find the total angular momentum of this particle. (b) Calculate L+ y (x, y, z) and (Y L+ W). (c)If a measurement of the z-component of the orbital angular momentum is carried out, find the probabilities corresponding to finding the results 0, h, and -h....
please help
1. The eigenfunctions of a particle in a square two-dimensional box with side lengths a = b = L are non, (x, y) = { sin ("T") sin (9,7%) = xn, (x)}n, (y) where n. (c) and on, (y) are one-dimensional particle-in-a-box wave functions in the x and y directions. a. Suppose we prepare the particle in such a way that it has a wave function V (2,y) given by 26,0) = Võru (s. 1) + Vedra ....
3.9. A particle of mass m is confined in the potential well 0 0<x < L oo elsewhere (a) At time t 0, the wave function for the particle is the one given in Problem 3.3. Calculate the probability that a measurement of the energy yields the value En, one of the allowed energies for a particle in the box. What are the numerical values for the probabilities of obtaining the ground-state energy E1 and the first-excited-state energy E2? Note:...
1. The wavefunction corresponding to Im> energy and angular momentum eigenstate of a particle rotating in a ring for m-l and m--1 are, respectively N2T where ? is the angular position of the particle relative to thex axis (see slide 15 of lecture 74a). (a) show that the probability density does not depend on 0. (b) Show that P,(o)-sin() where p, (0) rticle in the quantum state V, (d) p, (0) obviously resembles one of the orbitals of the is...
Problem 3 (Needed for Problem 4) A continuous random variable X is said to have an exponential distribution, written Exp(X), if its probability density function f is such that le- if > 0 10 if x < 0 f(0) = 0 where > 0 is a real number. 1. Compute the mean of X 2. Compute the variance of X 3. Compute the cumulative distribution function F of X. Use this to show that for any real numbers s and...
Problem 8.3 - A New Two-State System Consider a new two-level system with a Hamiltonian given by i = Ti 1461 – 12) (2) (3) Also consider an observable represented by the operator Ŝ = * 11/21 - *12/11: It should (hopefully) be clear that 1) and 2) are eigenkets of the Hamiltonian. Let $1) be an eigenket of S corresponding to the smaller eigenvalue of S and let S2) be an eigenket of S corresponding to the larger eigenvalue....
Suppose that the position of one particle at time t is given by Xi= 3 sint Yi = 2 cost 0 < t < 211 and the position of a second particle is given by X2 = -3+ cost y2 = 1+ sin t 0<t<21 (a)Graph the paths of both particles. How many points of ntersection are there? (b) Are any of these points of intersection collision points? In other words, are the particles ever at the same place at...
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