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3. Consider the wave function (x, t) = Ae-2 -ut Where A, 2, and are positive...
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. 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?
5. A free particle has the initial wave function, where A and a are positive real...
5. A free particle has the initial wave function, where A and a are positive real constants. (a) Normalize ψ(x,0). (b) Find φ(k). (c) Construct $(z,t), in the forn of an integral. (d) Discuss the limiting cases (a very large, and a very small).
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...
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...
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...
A. Normalize the wave function Ψ=Ae^(-ax^2) where A is the normalization constant and a is an...
A. Normalize the wave function Ψ=Ae^(-ax^2) where A is the normalization constant and a is an integer. A= ? B. What is the expected value of the momentum? <p> = ?
The most general wave function of a particle in the simple harmonic oscillator potential is: V(x,...
The most general wave function of a particle in the simple harmonic oscillator potential is: V(x, t) = (x)e-1st/ where and E, are the harmonic oscillator's stationary states and their corresponding energies. (a) Show that the expectation value of position is (hint: use the results of Problem 4): (v) = A cos (wt - ) where the real constants A and o are given by: 1 2 Ae-id-1 " Entichtin Interpret this result, comparing it with the motion of a...
Suppose at a certain time to the wave function is, Ψ(x,6) N for all x between...
Suppose at a certain time to the wave function is, Ψ(x,6) N for all x between the values ofx = 1 cm and x = 2 cm. For all values ofx outside the interval [12] the wave function is zero. a) Normalize the wave function. (Solve for N). Pay attention to units! b) Sketch the probability density V(x,/,)(x, as a function of x c) What is the probability of finding the electron between 1.5 cm and 2.0 cm? d) What...
Let's consider a function described in terms of its displacement y(x,t) at t 0 by: where a, b and e are positive constants a) Write an expression for this wave profile, having a speed in the nega...
Let's consider a function described in terms of its displacement y(x,t) at t 0 by: where a, b and e are positive constants a) Write an expression for this wave profile, having a speed in the negative x-direction, as a function of position and time (b) Sketch the profile of the wave at t-0 s and t 2 s if v1 m/s (c) Determine if the following functions describe a travelling wave: (i) vr,t) (ar+ bt c), where a, 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. 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?
5. A free particle has the initial wave function, where A and a are positive real constants. (a) Normalize ψ(x,0). (b) Find φ(k). (c) Construct $(z,t), in the forn of an integral. (d) Discuss the limiting cases (a very large, and a very small).
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...
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...
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...
The most general wave function of a particle in the simple harmonic oscillator potential is: V(x, t) = (x)e-1st/ where and E, are the harmonic oscillator's stationary states and their corresponding energies. (a) Show that the expectation value of position is (hint: use the results of Problem 4): (v) = A cos (wt - ) where the real constants A and o are given by: 1 2 Ae-id-1 " Entichtin Interpret this result, comparing it with the motion of a...
Suppose at a certain time to the wave function is, Ψ(x,6) N for all x between the values ofx = 1 cm and x = 2 cm. For all values ofx outside the interval [12] the wave function is zero. a) Normalize the wave function. (Solve for N). Pay attention to units! b) Sketch the probability density V(x,/,)(x, as a function of x c) What is the probability of finding the electron between 1.5 cm and 2.0 cm? d) What...
Let's consider a function described in terms of its displacement y(x,t) at t 0 by: where a, b and e are positive constants a) Write an expression for this wave profile, having a speed in the negative x-direction, as a function of position and time (b) Sketch the profile of the wave at t-0 s and t 2 s if v1 m/s (c) Determine if the following functions describe a travelling wave: (i) vr,t) (ar+ bt c), where a, b...
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