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fer pourice 5. Calculate the ground state expectation values for position, <x>, and momentum <p> for...
P7D.4 Calculate the expectation values of p, and p? for a particle in the state with...
P7D.4 Calculate the expectation values of p, and p? for a particle in the state with n- 2 in a one-dimensional square-well potential
calculate the expectation value of position x for a particle in a box of length L...
calculate the expectation value of position x for a particle in a box of length L in the state n=1
A. Momentum space We define the momentum space wave function φ(p) as where Ψ(x)is a solution...
A. Momentum space We define the momentum space wave function φ(p) as where Ψ(x)is a solution of the Schrödinger equation in configuration (position) space a) Show that the expectation values of and p can be written in terms of Ф(p) as <p(p)p(p)dp b) Demonstrate that φ(p) is normalized, ie if Ψ(x) is normalized. J ΙΨ(2)12dr-1 c) Show that Ф(p) 2dp can be interpreted as the probability to find a particle with momen tum between p and p+ dp
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...
64 Consider a particle in a one-dimensional box in the ground state v, and the first...
64 Consider a particle in a one-dimensional box in the ground state v, and the first excited state , described by the wave functions listed below. For each wave function, calculate the expec- tation value of the position (x), the expectation value of the position squared (), the expecta- tion value of the momentum (p), and the expectation value of the momentum squared (p2). 2 . 2x Ossa 0sxSa (b) Y2(x) = Vasin-
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...
7. What is the probability of finding a particle translating in the central third of a 1 dimensional box if it is in the (a) the ground state (b) the first excited state. (c) Compare these probab...
7. What is the probability of finding a particle translating in the central third of a 1 dimensional box if it is in the (a) the ground state (b) the first excited state. (c) Compare these probabiliies to the classical probability. (d) What is the average value for the position in the ground state? Do your answers make sense? 15P
7. What is the probability of finding a particle translating in the central third of a 1 dimensional box if...
Exercise 10.14 A particle is initially in its ground state in an infinite one-dimensional potential box...
Exercise 10.14 A particle is initially in its ground state in an infinite one-dimensional potential box with sides at x = 0 and x a. If the wall of the box at x-a is suddenly moved to x = 10a, calculate the probability of finding the particle in (a) the fourth excited (n = 5) state of the new box and (b) the ninth (n 10) excited state of the new box.
Q4. Consider the 1D infinite square-well potential shown in the figure below. V(x) O0 Position (a)...
Q4. Consider the 1D infinite square-well potential shown in the figure below. V(x) O0 Position (a) State the time-independent Schrödinger equation within the region 0<x<L for a particle with positive energy E 2 marks] (b) The wavefunction for 0<x< L can be written in the general form y(x) = Asin kx + B cos kx. Show that the normalised wavefunction for the 1D infinite potential well becomes 2sn'n? ?snT/where ( "1,2,3 ! where ( n = 1,2,5, ). [4 marks]...
5. Electron in an Infinite Potential Well a) Calculate the ground state and two next highest...
5. Electron in an Infinite Potential Well a) Calculate the ground state and two next highest energy levels for an electron confined to an infinitely high potential well of width l = 1.00E-10 m (roughly the diameter of a hydrogen atom in its ground state). b) If a photon were emitted when an electron jumps from n = 2 to n = 1, what would it's wavelength be? In which part of the spectrum does this lie?
P7D.4 Calculate the expectation values of p, and p? for a particle in the state with n- 2 in a one-dimensional square-well potential
A. Momentum space We define the momentum space wave function φ(p) as where Ψ(x)is a solution of the Schrödinger equation in configuration (position) space a) Show that the expectation values of and p can be written in terms of Ф(p) as <p(p)p(p)dp b) Demonstrate that φ(p) is normalized, ie if Ψ(x) is normalized. J ΙΨ(2)12dr-1 c) Show that Ф(p) 2dp can be interpreted as the probability to find a particle with momen tum between p and p+ dp
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...
64 Consider a particle in a one-dimensional box in the ground state v, and the first excited state , described by the wave functions listed below. For each wave function, calculate the expec- tation value of the position (x), the expectation value of the position squared (), the expecta- tion value of the momentum (p), and the expectation value of the momentum squared (p2). 2 . 2x Ossa 0sxSa (b) Y2(x) = Vasin-
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...
7. What is the probability of finding a particle translating in the central third of a 1 dimensional box if it is in the (a) the ground state (b) the first excited state. (c) Compare these probabiliies to the classical probability. (d) What is the average value for the position in the ground state? Do your answers make sense? 15P
7. What is the probability of finding a particle translating in the central third of a 1 dimensional box if...
Exercise 10.14 A particle is initially in its ground state in an infinite one-dimensional potential box with sides at x = 0 and x a. If the wall of the box at x-a is suddenly moved to x = 10a, calculate the probability of finding the particle in (a) the fourth excited (n = 5) state of the new box and (b) the ninth (n 10) excited state of the new box.
Q4. Consider the 1D infinite square-well potential shown in the figure below. V(x) O0 Position (a) State the time-independent Schrödinger equation within the region 0<x<L for a particle with positive energy E 2 marks] (b) The wavefunction for 0<x< L can be written in the general form y(x) = Asin kx + B cos kx. Show that the normalised wavefunction for the 1D infinite potential well becomes 2sn'n? ?snT/where ( "1,2,3 ! where ( n = 1,2,5, ). [4 marks]...
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