Consider a wave that is represented by ψ(x, t) = 4 cos (kx − ωt). where...
Consider a wave that is represented by ψ(x, t) = 4 cos (kx − ωt). where k = 2π/λ and ω = 2πf. The aim of the following exercises is to show that this expression captures many of the intuitive features of waves.
a) Consider a snapshot of the wave at t = 0. Use the expression to find the possible values of x at which the crests (maximum points) of the wave are located. By what distance are neighboring crests separated?
b) Show that the expression predicts that at any time t the value of ψ is the same for any two points separated by exactly one wavelength, i.e. show that ψ(x + λ, t) = ψ(x, t).
c) As time passes one can follow a particular crest by focusing on the argument of the cosine. For example suppose that kx − ωt = 0. This describes one particular crest of ψ. Where is this crest located at time ti = 0? Determine an expression for the location of this crest located at any later time tf . How far has the crest traveled during the time interval from ti to tf ? Determine an expression (in terms of k and ω) for the speed with which this crest travels.
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Consider a wave that is represented by ψ(x, t) = 4 cos (kx −
ωt). where...
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.
The magnetic field of an electromagnetic wave is described by By = B0cos(kx - ωt), where...
The magnetic field of an electromagnetic wave is described by By = B0cos(kx - ωt), where B0 = 3.5 × 10-6 T and ω = 2.5 × 107 rad/s. What is the amplitude of the corresponding electric field oscillations, E0, in terms of B0? What is the frequency of the electromagnetic wave, f, in terms of ω? What is the wavelength of the electromagnetic wave, λ, in terms of ω and the speed of light c?
Prove that E(x,t) = E0ei(kx-ωt) is a solution to the wave equation.
Prove that E(x,t) = E0ei(kx-ωt) is a solution to the wave equation.
9. 1.66 points Show that the wave function ψ-A ei(kx-at) is a solution to the Schrödinger equation, given below, where k-2π / λ and U-0. 2m dz2 Accomplish by calculating the following quantities. (Us...
9. 1.66 points Show that the wave function ψ-A ei(kx-at) is a solution to the Schrödinger equation, given below, where k-2π / λ and U-0. 2m dz2 Accomplish by calculating the following quantities. (Use the following as necessary: A, K, x, ,t, h, and m.) momentum Need Help?Read ItTalk to a Tutor
9. 1.66 points Show that the wave function ψ-A ei(kx-at) is a solution to the Schrödinger equation, given below, where k-2π / λ and U-0. 2m dz2 Accomplish...
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).
For the electromagnetic wave represented by the equations E_y(x, t) = E_max cos(kx + Wt), B_z(x,...
For the electromagnetic wave represented by the equations E_y(x, t) = E_max cos(kx + Wt), B_z(x, t) = -B_max cos(kx + omega t), find the direction of the Poynting vector. in the - y -direction in the +x -direction in the +y -direction in the -x -direction Part B Find the average magnitude of the Poynting vector. Express your answer in terms of the variables E_max, B_max, and appropriate constants (mu 0 or epsilon_0). submit
A transverse harmonic wave travels on a rope according to the following expression: y(x, t) =...
A transverse harmonic wave travels on a rope according to the
following expression:
y(x, t) = A cos(kx − ωt − φ)
The mass density of the rope is μ = 0.113 kg/m. x and y are
measured in meters and t in seconds. Using the graphs at of y vs t
at x=0 and y vs x at t=0 shown below, answer the following
questions:
(a) What is the value for the wave number, k, for the expression
y(x,t)?...
Consider the following wave function: y(x, t) = cos(kx - omega t). a. Show that the...
Consider the following wave function: y(x, t) = cos(kx - omega t). a. Show that the above function is an eigenfunction of the operator partialdifferential^2/partialdifferential x^2[...] and determine its eigenvalue. b. Show that the above function is a solution of the wave equation expressed as partialdifferential^2 y(x, t)/partialdifferential x^2 = 1/v^2 partialdifferential^2 y(x, t)/partialdifferential t^2, given the wave velocity is v = omega/k (where omega = 2 pi V and k = 2pi/lambda).
Sketch the profile of the wave (x,t) = A sin(kx-t+), where the initial phase is given...
Sketch the profile of the wave (x,t) = A sin(kx-t+), where
the initial phase is given by each of the following: =0 , =/2
and
3. (20 points) Sketch the profile of the wave P(x,t) = A sin(kx-ot+E), where the initial phase is given by each of the following: E=0, E=1/2 and <=n.
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?
9. 1.66 points Show that the wave function ψ-A ei(kx-at) is a solution to the Schrödinger equation, given below, where k-2π / λ and U-0. 2m dz2 Accomplish by calculating the following quantities. (Use the following as necessary: A, K, x, ,t, h, and m.) momentum Need Help?Read ItTalk to a Tutor
9. 1.66 points Show that the wave function ψ-A ei(kx-at) is a solution to the Schrödinger equation, given below, where k-2π / λ and U-0. 2m dz2 Accomplish...
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).
For the electromagnetic wave represented by the equations E_y(x, t) = E_max cos(kx + Wt), B_z(x, t) = -B_max cos(kx + omega t), find the direction of the Poynting vector. in the - y -direction in the +x -direction in the +y -direction in the -x -direction Part B Find the average magnitude of the Poynting vector. Express your answer in terms of the variables E_max, B_max, and appropriate constants (mu 0 or epsilon_0). submit
A transverse harmonic wave travels on a rope according to the
following expression:
y(x, t) = A cos(kx − ωt − φ)
The mass density of the rope is μ = 0.113 kg/m. x and y are
measured in meters and t in seconds. Using the graphs at of y vs t
at x=0 and y vs x at t=0 shown below, answer the following
questions:
(a) What is the value for the wave number, k, for the expression
y(x,t)?...
Consider the following wave function: y(x, t) = cos(kx - omega t). a. Show that the above function is an eigenfunction of the operator partialdifferential^2/partialdifferential x^2[...] and determine its eigenvalue. b. Show that the above function is a solution of the wave equation expressed as partialdifferential^2 y(x, t)/partialdifferential x^2 = 1/v^2 partialdifferential^2 y(x, t)/partialdifferential t^2, given the wave velocity is v = omega/k (where omega = 2 pi V and k = 2pi/lambda).
Sketch the profile of the wave (x,t) = A sin(kx-t+), where
the initial phase is given by each of the following: =0 , =/2
and
3. (20 points) Sketch the profile of the wave P(x,t) = A sin(kx-ot+E), where the initial phase is given by each of the following: E=0, E=1/2 and <=n.
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?
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