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Problem 2 Consider the wave function Where a, λ ω are positive constants. (a) Normalize (b)...
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...
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...
*Problem 1.6 Consider the Gaussian distribution A(x-a)2 where A, a, and λ are constants. (Look up...
*Problem 1.6 Consider the Gaussian distribution A(x-a)2 where A, a, and λ are constants. (Look up any integrals you need.) (a) Use Equation 1.16 to determine A. b) Find (x),(x-), and σ (c) Sketch the graph of ρ (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 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...
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...
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.
2. Consider the following 1-D wave equation with initial condition u (x, 0)- F (x) where F(x) is a given function. a) Show that u (x, t)-F (x - t) is a solution to the given PDE. b) If the function F...
2. Consider the following 1-D wave equation with initial condition u (x, 0)- F (x) where F(x) is a given function. a) Show that u (x, t)-F (x - t) is a solution to the given PDE. b) If the function F is given as 1; x< 10 x > 10 u(x, 0) = F(x) = use part (a) to write the solution u(x, t) c) Sketch u(x,0) and u(x,1) on the same u-versus-x graph d) Explain in your own...
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...
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...
*Problem 1.6 Consider the Gaussian distribution A(x-a)2 where A, a, and λ are constants. (Look up any integrals you need.) (a) Use Equation 1.16 to determine A. b) Find (x),(x-), and σ (c) Sketch the graph of ρ (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 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...
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...
2. Consider the following 1-D wave equation with initial condition u (x, 0)- F (x) where F(x) is a given function. a) Show that u (x, t)-F (x - t) is a solution to the given PDE. b) If the function F is given as 1; x< 10 x > 10 u(x, 0) = F(x) = use part (a) to write the solution u(x, t) c) Sketch u(x,0) and u(x,1) on the same u-versus-x graph d) Explain in your own...
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