Consider an electron moving along the positive x-axis in empty space (V(x) = 0), described by...
Consider an electron moving along the positive x-axis in empty space (V(x) = 0), described by the wave function ?(?) = sin(??) − ?cos(??)
Show that the following wave function is a solution to the Schrödinger Equation.
(− ℏ^2/2? ?^2/??2 + ?(?)) ?(?) = ??(?)
Homework Answers
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Consider an electron moving along the positive x-axis in empty
space (V(x) = 0), described by...
1. Consider an electron initially moving in the positive x-direction along the negative x-axis wi...
1. Consider an electron initially moving in the positive x-direction along the negative x-axis with an energy E. At the origin the potential energy U (x) changes abruptly, from U(x) = 0 for x < 0, to U(x) = 1.00 eV for x > 0. If E=1.02 eV, just higher than the barrier by 2%, what is the barrier penetration length? What is the reflectance? What is the transmittance? 2. Consider an electron initially moving in the positive x-direction along...
9. The wavefunction, p(x,t), of a particle moving along the x-axis, whose potential energy V(x) i...
Question #9 all parts thanks
9. The wavefunction, p(x,t), of a particle moving along the x-axis, whose potential energy V(x) is independent of time, is described by the one-dimensional non-relativistic Schrödinger equation (where m is its mass, h is the reduced Planck constant, i is the imaginary number): 2m (a) Verify that it is a parabolic equation (page E-1-2). [It has wave-like solutions, however.] (b) Use the substitution Px,t)-Xx)Tt) to separate the equation into two ODEs. (c) Solve for T,...
1) (35 points) The wave function for a particle moving along x axis between the limits 0 and L is: (x)-C sin (nx xL) where n are 1, 2, 3, A) Determine the normalization constant C B) Why can'...
1) (35 points) The wave function for a particle moving along x axis between the limits 0 and L is: (x)-C sin (nx xL) where n are 1, 2, 3, A) Determine the normalization constant C B) Why can't n take the value of 0, briefly explain C) For n-3 determine the values of x (in terms of L) that correspond to a maximum or a minimum in the wave function D) For n-3 determine the values of x (in...
Consider a plane-wave solution to the free Schrödinger equation (V = 0) in one space dimension,...
Consider a plane-wave solution to the free Schrödinger equation (V = 0) in one space dimension, with momentum pi. At time t = 0, the wavefunction takes the form Up (0,0) = P12/h. The lower index pı labels the momentum of this state. 1) What does the wavefunction look like at a later time t? 2) Next, consider another plane-wave state with a different momentum p2, with the wavefunction at t = 0 taking the form Up (2,0) = 1222/h....
Instead of assuming that a one-dimensional particle has no energy (v(x)=0), consider the case of a...
Instead of assuming that a one-dimensional
particle has no energy (v(x)=0), consider the case of a
one-dimensional particle which has finite, but constant, energy
V(x)= V sub zero.. Show that the ID particle in a box wave
functions. n(x)= A sin ((pi n x)/a). Also solve the Schrödinger
equation for this potential, and determine the energies En
Problem 2: Particle in a Box with Non-Zero Energy (2 points) Instead of assuming that a one-dimensional particle has no energy (V(x) =...
Question 21 Consider a free electron in one dimension (i.e. an electron free to move along...
Question 21 Consider a free electron in one dimension (i.e. an electron free to move along say the x-direction on (a) The time-independent Schrödinger equation is Αψη (x)-Εηψη (x), where is the Hamiltonian (total energy) operator, and ψη (x) are the electron wave functions associated with energies En Assuming the electron's energy entirely comprises kinetic energy (as it is 'free' there is no potential energy term), write down the Schrödinger equation given that the momentum operator in one- dimension is...
Consider a particle which is confined to move along the positive x-axis, and that has a...
Consider a particle which is confined to move along the positive x-axis, and that has a Hamiltonian where is a positive real constant having the dimensions of energy. Find the normalized wave function that corresponds to an energy eigenvalue of . The function should be finite everywhere along the positive x-axis and be square integrable. H = 8
1. The wave function describing a state of an electron confined to move along 2 the x axis is giv...
1. The wave function describing a state of an electron confined to move along 2 the x axis is given at time zero by W(x, 0)- Ae o2. Find the probability of finding the electron in a region dx centered at x-: σ. You need to first determine A and consider ơ as a known number.
1. The wave function describing a state of an electron confined to move along 2 the x axis is given at time zero by...
2. Consider an electron in a 1D potential box (V(x) = 0 for 0<x<L, V(x) =...
2. Consider an electron in a 1D potential box (V(x) = 0 for 0<x<L, V(x) = co otherwise) of length L = 1 nm. The electron is described by the wave function, c) = Jasin ( (a) Using the appropriate Hamiltonian derive an expression for the kinetic energy of the electron (5 marks) (b) Calculate the energy (in Joules) of the transition between the ground state and the 1 excited state. [3 marks]
Question 3 (1 point) A positive charge q is moving upward along the y-axis as shown...
Question 3 (1 point) A positive charge q is moving upward along the y-axis as shown in the diagram. What direction is the magnetic field produced by this charge at the point indicated on the x- axis? e X N O- Ot - cos oſ + sin bî sin di + cos oj
Question #9 all parts thanks
9. The wavefunction, p(x,t), of a particle moving along the x-axis, whose potential energy V(x) is independent of time, is described by the one-dimensional non-relativistic Schrödinger equation (where m is its mass, h is the reduced Planck constant, i is the imaginary number): 2m (a) Verify that it is a parabolic equation (page E-1-2). [It has wave-like solutions, however.] (b) Use the substitution Px,t)-Xx)Tt) to separate the equation into two ODEs. (c) Solve for T,...
1) (35 points) The wave function for a particle moving along x axis between the limits 0 and L is: (x)-C sin (nx xL) where n are 1, 2, 3, A) Determine the normalization constant C B) Why can't n take the value of 0, briefly explain C) For n-3 determine the values of x (in terms of L) that correspond to a maximum or a minimum in the wave function D) For n-3 determine the values of x (in...
Consider a plane-wave solution to the free Schrödinger equation (V = 0) in one space dimension, with momentum pi. At time t = 0, the wavefunction takes the form Up (0,0) = P12/h. The lower index pı labels the momentum of this state. 1) What does the wavefunction look like at a later time t? 2) Next, consider another plane-wave state with a different momentum p2, with the wavefunction at t = 0 taking the form Up (2,0) = 1222/h....
Instead of assuming that a one-dimensional
particle has no energy (v(x)=0), consider the case of a
one-dimensional particle which has finite, but constant, energy
V(x)= V sub zero.. Show that the ID particle in a box wave
functions. n(x)= A sin ((pi n x)/a). Also solve the Schrödinger
equation for this potential, and determine the energies En
Problem 2: Particle in a Box with Non-Zero Energy (2 points) Instead of assuming that a one-dimensional particle has no energy (V(x) =...
Question 21 Consider a free electron in one dimension (i.e. an electron free to move along say the x-direction on (a) The time-independent Schrödinger equation is Αψη (x)-Εηψη (x), where is the Hamiltonian (total energy) operator, and ψη (x) are the electron wave functions associated with energies En Assuming the electron's energy entirely comprises kinetic energy (as it is 'free' there is no potential energy term), write down the Schrödinger equation given that the momentum operator in one- dimension is...
1. The wave function describing a state of an electron confined to move along 2 the x axis is given at time zero by W(x, 0)- Ae o2. Find the probability of finding the electron in a region dx centered at x-: σ. You need to first determine A and consider ơ as a known number.
1. The wave function describing a state of an electron confined to move along 2 the x axis is given at time zero by...
2. Consider an electron in a 1D potential box (V(x) = 0 for 0<x<L, V(x) = co otherwise) of length L = 1 nm. The electron is described by the wave function, c) = Jasin ( (a) Using the appropriate Hamiltonian derive an expression for the kinetic energy of the electron (5 marks) (b) Calculate the energy (in Joules) of the transition between the ground state and the 1 excited state. [3 marks]
Question 3 (1 point) A positive charge q is moving upward along the y-axis as shown in the diagram. What direction is the magnetic field produced by this charge at the point indicated on the x- axis? e X N O- Ot - cos oſ + sin bî sin di + cos oj
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