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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...
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.,...
3.9. A particle of mass m is confined in the potential well 0 0<x < L...
3.9. A particle of mass m is confined in the potential well 0 0<x < L oo elsewhere (a) At time t 0, the wave function for the particle is the one given in Problem 3.3. Calculate the probability that a measurement of the energy yields the value En, one of the allowed energies for a particle in the box. What are the numerical values for the probabilities of obtaining the ground-state energy E1 and the first-excited-state energy E2? Note:...
Answers can be more than one: VII. (12pts) Consider the following potential energy: region 1: U(X)...
Answers can be more than one:
VII. (12pts) Consider the following potential energy: region 1: U(X) = U. x < 0 region 2: U(X) = 0 0<x</ Uo region 3: U(x) = U. x>L ТЕ where U. >0. We want to consider a particle with energy E such that 0 < E<Uo. There are two possible forms for the wave function that might be used to represent the particle: (x) = 4 sin kyx+ B, coskx v(x) = 4e** +...
please explain all, thanks! 4. (60 pts) A particle in an infinite square well of width...
please explain all, thanks!
4. (60 pts) A particle in an infinite square well of width L has an initial wave function (x,t = 0) = Ax(L - x)2, OSX SL a) Find y(x, t) fort > 0. You first have to normalize the wave function. Hint: this is best expressed an infinite series: show that the wave function coefficients are on = * 31% (12 – n?)(1-(-1)") → (n = 87315 (12 - nºre?); n odd. b) Which energy...
Please answer the question in full and show all work. We have seen that the absolute...
Please answer the question in full and show all work.
We have seen that the absolute square of the wave function VI,t) can be interpreted as the probability density for the location of the particle at time t. We have also seen that a particle's quantum state can be represented as a linear combination of eigenstates of a physical observable Q: V) SIT) where Q n ) = qn|n) and represents the probability to find the particle in the eigenstate...
Question 2: A particle of mass m moves in a potential energy U(x) that is zero...
Question 2: A particle of mass m moves in a potential energy U(x) that is zero forェ* 0 and is-oo at r-0. This is am attractive delta function, very odd. Do not worry about the physical meaning of the potential, just roll with it for now. The system is described by the wave function Afor <0 where a is a real, positive constant with dimensions of 1/Length, and A is the normalization constant, treat it as a unknown complex-number for...
Please make the hand writing legible. Thanks Consider the situation depicted below, in which an incompressible...
Please make the hand writing legible. Thanks
Consider the situation depicted below, in which an incompressible fluid flows over a flat surface of solid. Upstream of the surface, the fluid has velocity U and uniform temperature To. As the fluid is viscous, both a momentum boundary layer, and a thermal boundary layer form, and heat is transferred to the solid surface. A convective coefficient h can be used to describe the dimensional heat transfer rate to the solid, and is...
2.2 Two-level system A particle in the box is described by the following wavefunction 1 1...
2.2 Two-level system A particle in the box is described by the following wavefunction 1 1 V(x, t) + V2 V2 = Um(x)e -i(Em/h) In other words, this state is a superposition of two modes: n-th, and m-th. A superposition that involves only two modes (not necessarily particle in the box modes, but any two modes) is called a "two-level system”. A more modern name for such a superposition is a "qubit”. a) Come up with an expression for the...
A particle of mass m is bound by the spherically-symmetric three-dimensional harmonic- oscillator potential energy ,...
A particle of mass m is bound by the spherically-symmetric three-dimensional harmonic- oscillator potential energy , and ф are the usual spherical coordinates. (a) In the form given above, why is it clear that the potential energy function V) is (b) For this problem, it will be more convenient to express this spherically-symmetric where r , spherically symmetric? A brief answer is sufficient. potential energy in Cartesian coordinates x, y, and z as physically the same potential energy as the...
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...
3.9. A particle of mass m is confined in the potential well 0 0<x < L oo elsewhere (a) At time t 0, the wave function for the particle is the one given in Problem 3.3. Calculate the probability that a measurement of the energy yields the value En, one of the allowed energies for a particle in the box. What are the numerical values for the probabilities of obtaining the ground-state energy E1 and the first-excited-state energy E2? Note:...
Answers can be more than one:
VII. (12pts) Consider the following potential energy: region 1: U(X) = U. x < 0 region 2: U(X) = 0 0<x</ Uo region 3: U(x) = U. x>L ТЕ where U. >0. We want to consider a particle with energy E such that 0 < E<Uo. There are two possible forms for the wave function that might be used to represent the particle: (x) = 4 sin kyx+ B, coskx v(x) = 4e** +...
please explain all, thanks!
4. (60 pts) A particle in an infinite square well of width L has an initial wave function (x,t = 0) = Ax(L - x)2, OSX SL a) Find y(x, t) fort > 0. You first have to normalize the wave function. Hint: this is best expressed an infinite series: show that the wave function coefficients are on = * 31% (12 – n?)(1-(-1)") → (n = 87315 (12 - nºre?); n odd. b) Which energy...
Please answer the question in full and show all work.
We have seen that the absolute square of the wave function VI,t) can be interpreted as the probability density for the location of the particle at time t. We have also seen that a particle's quantum state can be represented as a linear combination of eigenstates of a physical observable Q: V) SIT) where Q n ) = qn|n) and represents the probability to find the particle in the eigenstate...
Question 2: A particle of mass m moves in a potential energy U(x) that is zero forェ* 0 and is-oo at r-0. This is am attractive delta function, very odd. Do not worry about the physical meaning of the potential, just roll with it for now. The system is described by the wave function Afor <0 where a is a real, positive constant with dimensions of 1/Length, and A is the normalization constant, treat it as a unknown complex-number for...
Please make the hand writing legible. Thanks
Consider the situation depicted below, in which an incompressible fluid flows over a flat surface of solid. Upstream of the surface, the fluid has velocity U and uniform temperature To. As the fluid is viscous, both a momentum boundary layer, and a thermal boundary layer form, and heat is transferred to the solid surface. A convective coefficient h can be used to describe the dimensional heat transfer rate to the solid, and is...
2.2 Two-level system A particle in the box is described by the following wavefunction 1 1 V(x, t) + V2 V2 = Um(x)e -i(Em/h) In other words, this state is a superposition of two modes: n-th, and m-th. A superposition that involves only two modes (not necessarily particle in the box modes, but any two modes) is called a "two-level system”. A more modern name for such a superposition is a "qubit”. a) Come up with an expression for the...
A particle of mass m is bound by the spherically-symmetric three-dimensional harmonic- oscillator potential energy , and ф are the usual spherical coordinates. (a) In the form given above, why is it clear that the potential energy function V) is (b) For this problem, it will be more convenient to express this spherically-symmetric where r , spherically symmetric? A brief answer is sufficient. potential energy in Cartesian coordinates x, y, and z as physically the same potential energy as the...
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