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3. Consider a free particle of mass m and energy E moving in three dimensions. What...
[12 6. Consid er a particle of mass m moving in an infinitely deep square well...
[12 6. Consid er a particle of mass m moving in an infinitely deep square well potential of width a, whose wave function at time t 0 is where on Ce) is the normaized wave function of the n-th eigenstate of the Hamitonian of that particle The corresponding eigen-energy of the n-th state is 2ma?n 1,2,3,... (e) Find the average energy of the system (ie. the expectation value () (b) Write down the wave function p(z,t) at a later time...
Consider a particle of mass m moving in a one-dimensional potential of the form V. for...
Consider a particle of mass m moving in a one-dimensional potential of the form V. for 0<x<b, V(a) = 0 for Islal<e, for 1212, with V., b and c positive constants and c>b. a Explain why the wave function of the particle can be assumed to be cither an even function or an odd function of a. b For the case that the energy E of the particle is in the range 0<ESV., find the (unnormalized) even cigenfunctions and give...
A particle of mass m and energy E moving along the x axis is subjected to a potential energy function U(x).
\((25\) marks) A particle of mass \(m\) and energy \(E\) moving along the \(x\) axis is subjected to a potential energy function \(U(x) .\) (a) Suppose \(\psi_{1}(x)\) and \(\psi_{2}(\mathrm{x})\) are two wave functions of the system with the same energy \(E .\) Derive an expression to relate \(\psi_{1}(x), \psi_{2}(x)\), and their derivatives. (b) By requiring the wave functions to vanish at infinity, show that \(\psi_{1}(x)\) and \(\psi_{2}(x)\) can at most differ by a multiplicative constant. Hence, what conclusion can you...
1. A free particle of mass m moving from the left in one dimension scatters from...
1. A free particle of mass m moving from the left in one dimension scatters from the potential V(x) αδ(x). Suppose that the wave number of the particle is k and that α > 0. a. State the general form of the wave function including reflected and transmitted waves. b. Find the amplitude t of the transmitted wave in terms of α, k, m, and h. Find T
3. Consider a particle of mass m moving in a potential given by: W (2, y,...
3. Consider a particle of mass m moving in a potential given by: W (2, y, z) = 0 < x <a,0 < y <a l+o, elsewhere a) Write down the total energy and the 3D wavefunction for this particle. b) Assuming that hw > 312 h2/(2ma), find the energies and the corresponding degen- eracies for the ground state and the first excited state. c) Assume now that, in addition to the potential V(x, y, z), this particle also has...
What would be the result of a kinetic energy measurement on a free quantum particle? (i.e....
What would be the result of a kinetic energy measurement on a free quantum particle? (i.e. potential energy V(x) = 0) of mass m with a wave-function ψ(x) = e^(-x^2) A hint for this question: Consider only the kinetic energy operator. Is the given function an eigenfunction of this operator? If yes, what will be the result of the measurement? If not an eigenfunction, what would be the result of the measurement?
At time t = 0, a mass-m particle in a one-dimensional potential well is in a...
At time t = 0, a mass-m particle in a one-dimensional potential well is in a state given by the normalised wave function (x, 0) =3/2eAl2| | -ao x << 0, realU>0. Find the potential energy V = the energy eigenvalue E. Fix zero energy according to the convention V(x) » 0 for ao. Is there a delta function singularity at x0? V (x) for which this is an energy eigenstate and determine [6]
At time t = 0, a...
Scattering #1 Consider the "downstep" potential shown. A particle of mass m and energy E, inciden...
Scattering #1 Consider the "downstep" potential shown. A particle of mass m and energy E, incident from the left, strikes a potential energy drop-off of depth Vo 0 (2 pts) Using classical physics, consider a particle incident with speed vo. Use conservation of energy to find the speed on the right vf. ALSO, what is the probability that a given particle will "transmit" from the left side to the right side (again, classically)? A. B. (4 pts) This problem is...
Help with question (e) 3. Wave functions. For this question, suppose a particle of mass m...
Help with question (e)
3. Wave functions. For this question, suppose a particle of mass m is described by the wave function 23ha Axe-ax2c"Tat Ψ(x, t) (a) Show that the wave function above is a solution to the full time dependent Schrödinger equation if the potential energy function is given by 2h2a2 U(x)2 (b) What are the dimensional units of a and A in the wave function? (c) Show that the probability distribution associated with V(r, r) is independent of...
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...
[12 6. Consid er a particle of mass m moving in an infinitely deep square well potential of width a, whose wave function at time t 0 is where on Ce) is the normaized wave function of the n-th eigenstate of the Hamitonian of that particle The corresponding eigen-energy of the n-th state is 2ma?n 1,2,3,... (e) Find the average energy of the system (ie. the expectation value () (b) Write down the wave function p(z,t) at a later time...
Consider a particle of mass m moving in a one-dimensional potential of the form V. for 0<x<b, V(a) = 0 for Islal<e, for 1212, with V., b and c positive constants and c>b. a Explain why the wave function of the particle can be assumed to be cither an even function or an odd function of a. b For the case that the energy E of the particle is in the range 0<ESV., find the (unnormalized) even cigenfunctions and give...
1. A free particle of mass m moving from the left in one dimension scatters from the potential V(x) αδ(x). Suppose that the wave number of the particle is k and that α > 0. a. State the general form of the wave function including reflected and transmitted waves. b. Find the amplitude t of the transmitted wave in terms of α, k, m, and h. Find T
3. Consider a particle of mass m moving in a potential given by: W (2, y, z) = 0 < x <a,0 < y <a l+o, elsewhere a) Write down the total energy and the 3D wavefunction for this particle. b) Assuming that hw > 312 h2/(2ma), find the energies and the corresponding degen- eracies for the ground state and the first excited state. c) Assume now that, in addition to the potential V(x, y, z), this particle also has...
At time t = 0, a mass-m particle in a one-dimensional potential well is in a state given by the normalised wave function (x, 0) =3/2eAl2| | -ao x << 0, realU>0. Find the potential energy V = the energy eigenvalue E. Fix zero energy according to the convention V(x) » 0 for ao. Is there a delta function singularity at x0? V (x) for which this is an energy eigenstate and determine [6]
At time t = 0, a...
Scattering #1 Consider the "downstep" potential shown. A particle of mass m and energy E, incident from the left, strikes a potential energy drop-off of depth Vo 0 (2 pts) Using classical physics, consider a particle incident with speed vo. Use conservation of energy to find the speed on the right vf. ALSO, what is the probability that a given particle will "transmit" from the left side to the right side (again, classically)? A. B. (4 pts) This problem is...
Help with question (e)
3. Wave functions. For this question, suppose a particle of mass m is described by the wave function 23ha Axe-ax2c"Tat Ψ(x, t) (a) Show that the wave function above is a solution to the full time dependent Schrödinger equation if the potential energy function is given by 2h2a2 U(x)2 (b) What are the dimensional units of a and A in the wave function? (c) Show that the probability distribution associated with V(r, r) is independent of...
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...
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