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5. A free particle has the initial wave function, where A and a are positive real...
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.,...
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.
3 Rockin' in the Free World Consider a free particle whose state at time t 0...
3 Rockin' in the Free World Consider a free particle whose state at time t 0 is given by a gaussian wave packet, a2z2 V(x,0)Ae- for real constants A, a 1. Normalize V(x,0), i.e., find A. 2. Find Ψ(x, t). You can do the integral by completing the square in the exponent to get it into the form of a gaussian
3. At time t-0 a particle is represented by the wave function A-if 0 < x<a...
3. At time t-0 a particle is represented by the wave function A-if 0 < x<a ψ(x,0) = 0 otherwise where A, a, and b are constants. a) Normalize ψ(x,0). b) Draw (x,0). c) Where is the particle most likely to be found at t-0? d) What is the probability of finding the particle to the left of a? e) What is the expectation value of x?
0 is given by a gaussian wave packet Consider a free particle whose state at time...
0 is given by a gaussian wave packet Consider a free particle whose state at time t (x, 0) Ae2/a2 for real constants A, a. (a) Normalize (r, 0), i.e., find A (b) Find (r, t). You can do the integral by completing the square in the exponent to get it into the form of a gaussian (c) Compute the probability density (, t), expressing your answer in terms of the quantity w av1(2ht/ma2)2 Sketch the probability density as a...
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...
Problem 2: Time development of a free particle The wave function of a free particle at time t- 0 is given by exp(2K1T N...
Problem 2: Time development of a free particle The wave function of a free particle at time t- 0 is given by exp(2K1T Now answer the questions below. l. what is the time evolved wave function ψ(z,t) ? points 2. What is the average momentum at any future time? 4 points 3. What is the average energy at any future time ? 3 points
Problem 2: Time development of a free particle The wave function of a free particle at...
A free proton has a wave function Psi (x) = A sin (kx), where k =...
A free proton has a wave function Psi (x) = A sin (kx), where k = 1.2 times 10^10 m^-1 What is the proton's lambda? What is the proton's momentum? What is the proton's speed? Normalize Psi (x) if the wave only exists inside an infinite square well with width a = 2.1 m, (so that Psi (x) = A sin (kx) between 0 < x < a and Psi (x) = 0 otherwise).
3. Consider the wave function (x, t) = Ae-2 -ut Where A, 2, and are positive...
3. Consider the wave function (x, t) = Ae-2 -ut Where A, 2, and are positive real constants. (a) Normalize Y. (b) Determine the expectation values of x and x?. (c) Find the standard deviation of x. Sketch the graph of V', as a function of x, and mark the points (x) + a) and (x) -o to illustrate the sense in which represents the spread" in x. What is the probability that the particle would be found outside this...
Problem 2 Consider the wave function Where a, λ ω are positive constants. (a) Normalize (b)...
Problem 2 Consider the wave function Where a, λ ω are positive constants. (a) Normalize (b) Determine the expectation values ofx and x; (c) Find the standard deviation ofx. Sketch the graph of 1992, as a function ofx, and mark the points (<x> + σ) and 〈X>-07, to illustrate the sense in which σ represents the "spread" in x, what is the probability that the particle would be found outside this range?
3 Rockin' in the Free World Consider a free particle whose state at time t 0 is given by a gaussian wave packet, a2z2 V(x,0)Ae- for real constants A, a 1. Normalize V(x,0), i.e., find A. 2. Find Ψ(x, t). You can do the integral by completing the square in the exponent to get it into the form of a gaussian
3. At time t-0 a particle is represented by the wave function A-if 0 < x<a ψ(x,0) = 0 otherwise where A, a, and b are constants. a) Normalize ψ(x,0). b) Draw (x,0). c) Where is the particle most likely to be found at t-0? d) What is the probability of finding the particle to the left of a? e) What is the expectation value of x?
0 is given by a gaussian wave packet Consider a free particle whose state at time t (x, 0) Ae2/a2 for real constants A, a. (a) Normalize (r, 0), i.e., find A (b) Find (r, t). You can do the integral by completing the square in the exponent to get it into the form of a gaussian (c) Compute the probability density (, t), expressing your answer in terms of the quantity w av1(2ht/ma2)2 Sketch the probability density as a...
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...
Problem 2: Time development of a free particle The wave function of a free particle at time t- 0 is given by exp(2K1T Now answer the questions below. l. what is the time evolved wave function ψ(z,t) ? points 2. What is the average momentum at any future time? 4 points 3. What is the average energy at any future time ? 3 points
Problem 2: Time development of a free particle The wave function of a free particle at...
A free proton has a wave function Psi (x) = A sin (kx), where k = 1.2 times 10^10 m^-1 What is the proton's lambda? What is the proton's momentum? What is the proton's speed? Normalize Psi (x) if the wave only exists inside an infinite square well with width a = 2.1 m, (so that Psi (x) = A sin (kx) between 0 < x < a and Psi (x) = 0 otherwise).
3. Consider the wave function (x, t) = Ae-2 -ut Where A, 2, and are positive real constants. (a) Normalize Y. (b) Determine the expectation values of x and x?. (c) Find the standard deviation of x. Sketch the graph of V', as a function of x, and mark the points (x) + a) and (x) -o to illustrate the sense in which represents the spread" in x. What is the probability that the particle would be found outside this...
Problem 2 Consider the wave function Where a, λ ω are positive constants. (a) Normalize (b) Determine the expectation values ofx and x; (c) Find the standard deviation ofx. Sketch the graph of 1992, as a function ofx, and mark the points (<x> + σ) and 〈X>-07, to illustrate the sense in which σ represents the "spread" in x, what is the probability that the particle would be found outside this range?
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