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GROUP WORK The wave function of a particle with mass m is 4(x, e) = a...
part (e),(f) and (g) 3. Wave functions (40 marks) Consider particle described by wave function (x)...
part (e),(f) and (g)
3. Wave functions (40 marks) Consider particle described by wave function (x) = Ce-x/a for x > 0, and otherwise (C is a real and non-negative). (a) Normalise (x) and plot it (you can use a computer to plotit). (b) Calculate the probability that the particle is located within distance a from the origin. (e) Find mean value of position measurement. (d) Find mean value of momentum measurement. Hint: use the fact that (x) is purely...
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.,...
Extra Credit (3 points to Mideterm-2) Q1. A particle is described by the wave function (x)...
Extra Credit (3 points to Mideterm-2) Q1. A particle is described by the wave function (x) b(a2-x2) for -a sx s a and (x) 0 for x -a and x +a, where a and b are positive real constants. (a) Using the normalization condition, find b in terms a. (b) What is the probability to find the particle at x = +a/2 in a small interval ofwidth 0.01 a ? (c) What is the probability for the particle to be...
Consider a particle of mass m that is described by the wave function (x, t) =...
Consider a particle of mass m that is described by the wave function (x, t) = Ce~iwte-(x/l)2 where C and l are real and positive constants, with / being the characteristic length-scale in the problem Calculate the expectation values of position values of 2 and p2. and momentum p, as well as the expectation Find the standard deviations O and op. Are they consistent with the uncertainty principle? to be independent What should be the form of the potential energy...
A particle has a rest mass of 6.95×10^-27 and a momentum of 4.73×10^−18 kg⋅m/s. Determine the...
A particle has a rest mass of 6.95×10^-27 and a momentum of 4.73×10^−18 kg⋅m/s. Determine the total relativistic energy of the particle. E = ______________ J Find the ratio of the particle's relativistic kinetic energy to its rest energy. ?? rest = _________________
7. The kinetic energy, k, of a particle of mass m is given below, where the...
7. The kinetic energy, k, of a particle of mass m is given below, where the velocity, v, of the particle is constrained to [-1,1] Suppose that a particular particle is known to have mass m - 2 and that the probability that its velocity is in [a,b] is given below. Let K denote the random variable that characterizes the particle's kinetic energy. What is the probability that the kinetic energy is greater than one half? That is, find P[K...
Please show all work with algebra. 7 6. A particle with a mass of 0.500 kg...
Please show all work with algebra.
7 6. A particle with a mass of 0.500 kg is attached to a horizontal spring with a force 1.00 m and has a speed of constant of 50.0 N/m. Att the particle is at 10V3 m/s to the right. (a) (8 points) Find the amplitude and the initial phase of the oscillation. (b) (7 points) What is is the period of the oscillation? Find the length of a simple pendulum with the same...
Problem 1. Wave function An electron is described by a wave function: for x < 0...
Problem 1. Wave function An electron is described by a wave function: for x < 0 *(z) = { ce Ce-s/1(1 – e-3/4) for x > 0 : where I is a constant length, and C is the normalization constant. 1. Find C. 2. Where an electron is most likely to be found; that is, for what value of x is the prob: bility for finding electron largest? 3. What is the average coordinate 7 of the electron? 4. What...
Consider a particle confined to one dimension and positive with the wave function Nxear, x20 x<0...
Consider a particle confined to one dimension and positive with the wave function Nxear, x20 x<0 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: a) Find the normalization constant N. What are the units of your result and do they make sense? b) What is the most probable location to find the particle, or more precisely, at what z...
A particle is described by the (non-normalized) wave function ψ(x) = (a^2 − x^2) for −a...
A particle is described by the (non-normalized) wave function ψ(x) = (a^2 − x^2) for −a ≤ x ≤ +a and ψ(x) = 0 for x ≤ −a and x ≥ +a, where a is a positive real constant. The probability that the particle is found between x = +a/2 and x = a. Calculate the values of the expectation value of momentum <p> and the standard deviation of momentum σp.
part (e),(f) and (g)
3. Wave functions (40 marks) Consider particle described by wave function (x) = Ce-x/a for x > 0, and otherwise (C is a real and non-negative). (a) Normalise (x) and plot it (you can use a computer to plotit). (b) Calculate the probability that the particle is located within distance a from the origin. (e) Find mean value of position measurement. (d) Find mean value of momentum measurement. Hint: use the fact that (x) is purely...
Extra Credit (3 points to Mideterm-2) Q1. A particle is described by the wave function (x) b(a2-x2) for -a sx s a and (x) 0 for x -a and x +a, where a and b are positive real constants. (a) Using the normalization condition, find b in terms a. (b) What is the probability to find the particle at x = +a/2 in a small interval ofwidth 0.01 a ? (c) What is the probability for the particle to be...
Consider a particle of mass m that is described by the wave function (x, t) = Ce~iwte-(x/l)2 where C and l are real and positive constants, with / being the characteristic length-scale in the problem Calculate the expectation values of position values of 2 and p2. and momentum p, as well as the expectation Find the standard deviations O and op. Are they consistent with the uncertainty principle? to be independent What should be the form of the potential energy...
7. The kinetic energy, k, of a particle of mass m is given below, where the velocity, v, of the particle is constrained to [-1,1] Suppose that a particular particle is known to have mass m - 2 and that the probability that its velocity is in [a,b] is given below. Let K denote the random variable that characterizes the particle's kinetic energy. What is the probability that the kinetic energy is greater than one half? That is, find P[K...
Please show all work with algebra.
7 6. A particle with a mass of 0.500 kg is attached to a horizontal spring with a force 1.00 m and has a speed of constant of 50.0 N/m. Att the particle is at 10V3 m/s to the right. (a) (8 points) Find the amplitude and the initial phase of the oscillation. (b) (7 points) What is is the period of the oscillation? Find the length of a simple pendulum with the same...
Problem 1. Wave function An electron is described by a wave function: for x < 0 *(z) = { ce Ce-s/1(1 – e-3/4) for x > 0 : where I is a constant length, and C is the normalization constant. 1. Find C. 2. Where an electron is most likely to be found; that is, for what value of x is the prob: bility for finding electron largest? 3. What is the average coordinate 7 of the electron? 4. What...
Consider a particle confined to one dimension and positive with the wave function Nxear, x20 x<0 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: a) Find the normalization constant N. What are the units of your result and do they make sense? b) What is the most probable location to find the particle, or more precisely, at what z...
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