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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...
3. Consider a free particle on a circle. That is, consider V(z) = 0 and wave...
3. Consider a free particle on a circle. That is, consider V(z) = 0 and wave functions Ψ(z, t) which are periodic functions of z: Ψ(z,t) = Ψ(z + L, t). a) Solve the Time-Independent Schroedinger equation. For each allowed energy, En, you will find two solutions, (s). Why does this not contradict the theorem that we proved in class about the non-degeneracy of the solutions to the TISE in one dimension? b) Start with the initial condition Ψ(z,0) sin2(nz/L)....
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?
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.,...
4 At a given time, the wave function of a particle of mass m moving in...
4 At a given time, the wave function of a particle of mass m moving in a potential well is constant}(xyz) exp(-aa real a > 0. Calculate the probabilities that measurements of L2 and L,z yield the results 2h2 and 0 respectively. [6]
4 At a given time, the wave function of a particle of mass m moving in a potential well is constant}(xyz) exp(-aa real a > 0. Calculate the probabilities that measurements of L2 and L,z yield the...
The one-dimensional wave function for a particle over all space... may be exp ressed as a) Apply the momentum and energy Operators to ψ ( ie, p Ψ & ΕΨ ) as to verify the following pshk and Es...
The one-dimensional wave function for a particle over all space... may be exp ressed as a) Apply the momentum and energy Operators to ψ ( ie, p Ψ & ΕΨ ) as to verify the following pshk and Eshω Schrodinger sequation...-Nay equation... Ew andthen wufythefollowing: b) Substitute w into 2m ax E-Pi 2m
The one-dimensional wave function for a particle over all space... may be exp ressed as a) Apply the momentum and energy Operators to ψ ( ie, p...
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...
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
In a one-dimensional system at time t-0, the wave function of a particle is given by...
In a one-dimensional system at time t-0, the wave function of a particle is given by the function xfor 0SxSL 0 elsewhere -A opl as sketched in the diagram, where A is a positive constant. If the position of the particle is measured at time t-0, what is the probability of finding it somewhere in the interval 0 sx S L22 Specify your answer as a fraction or as a decimal correct to 2 significant figures. probability
Problem 3: A free particle of mass m in one dimension is in the state Hbr...
Problem 3: A free particle of mass m in one dimension is in the state Hbr Ψ(z, t = 0) = Ae-ar with A, a and b real positive constants. a) Calculate A by normalizing v. b) Calculate the expectation values of position and momentum of the particle at t 0 c) Calculate the uncertainties ΔΧ and Δ1) for the position and momentum at t 0, Do they satisfy the Heisenberg relation? d) Find the wavefunction Ψ(x, t) at a...
A free electron has a wave function ψ(x)= Asin (5x1010 x) where x is measured in...
A free electron has a wave function ψ(x)= Asin (5x1010 x) where x is measured in meters. Find the electron's de Broglie wavelength the electron's momentum a. b, 3. When an electron is confined in the semi-infinite square, its wave function will be in the form Asin kx for0<x<L ψ(x)- Ce for x> L having L = 5 nm and k = 1.7 / nm. a. Find the energy of the state. b. Write down the matching conditions that the...
3. Consider a free particle on a circle. That is, consider V(z) = 0 and wave functions Ψ(z, t) which are periodic functions of z: Ψ(z,t) = Ψ(z + L, t). a) Solve the Time-Independent Schroedinger equation. For each allowed energy, En, you will find two solutions, (s). Why does this not contradict the theorem that we proved in class about the non-degeneracy of the solutions to the TISE in one dimension? b) Start with the initial condition Ψ(z,0) sin2(nz/L)....
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?
4 At a given time, the wave function of a particle of mass m moving in a potential well is constant}(xyz) exp(-aa real a > 0. Calculate the probabilities that measurements of L2 and L,z yield the results 2h2 and 0 respectively. [6]
4 At a given time, the wave function of a particle of mass m moving in a potential well is constant}(xyz) exp(-aa real a > 0. Calculate the probabilities that measurements of L2 and L,z yield the...
The one-dimensional wave function for a particle over all space... may be exp ressed as a) Apply the momentum and energy Operators to ψ ( ie, p Ψ & ΕΨ ) as to verify the following pshk and Eshω Schrodinger sequation...-Nay equation... Ew andthen wufythefollowing: b) Substitute w into 2m ax E-Pi 2m
The one-dimensional wave function for a particle over all space... may be exp ressed as a) Apply the momentum and energy Operators to ψ ( ie, p...
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...
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
In a one-dimensional system at time t-0, the wave function of a particle is given by the function xfor 0SxSL 0 elsewhere -A opl as sketched in the diagram, where A is a positive constant. If the position of the particle is measured at time t-0, what is the probability of finding it somewhere in the interval 0 sx S L22 Specify your answer as a fraction or as a decimal correct to 2 significant figures. probability
Problem 3: A free particle of mass m in one dimension is in the state Hbr Ψ(z, t = 0) = Ae-ar with A, a and b real positive constants. a) Calculate A by normalizing v. b) Calculate the expectation values of position and momentum of the particle at t 0 c) Calculate the uncertainties ΔΧ and Δ1) for the position and momentum at t 0, Do they satisfy the Heisenberg relation? d) Find the wavefunction Ψ(x, t) at a...
A free electron has a wave function ψ(x)= Asin (5x1010 x) where x is measured in meters. Find the electron's de Broglie wavelength the electron's momentum a. b, 3. When an electron is confined in the semi-infinite square, its wave function will be in the form Asin kx for0<x<L ψ(x)- Ce for x> L having L = 5 nm and k = 1.7 / nm. a. Find the energy of the state. b. Write down the matching conditions that the...
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