Suppose a particle has zero potential energy for x < 0, a constant value V, for...
Suppose a particle has zero potential energy for x < 0, a constant value V, for 0 ≤ x ≤ L, and then zero for x > L. Sketch the potential. Now suppose that wavefunction is a sine wave on the left of the barrier, declines exponentially inside the barrier, and then becomes a sine wave on the right, being continuous everywhere. Sketch the wavefunction on your sketch of the potential energy.
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Suppose a particle has zero potential energy for x
< 0, a constant value V, for...
Suppose a particle has zero potential energy for x < 0, a constant value V, for...
Suppose a particle has zero potential energy for x < 0, a constant value V, for 0 ≤ x ≤ L, and then zero for x > L. Sketch the potential. Now suppose that wavefunction is a sine wave on the left of the barrier, declines exponentially inside the barrier, and then becomes a sine wave on the right, being continuous everywhere. Sketch the wavefunction on your sketch of the potential energy.
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...
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) =...
(III) Quantum Tunneling Consider an electron in 1D in presence of a potential barrier of width...
(III) Quantum Tunneling Consider an electron in 1D in presence of a potential barrier of width L represented by a step function ſo I<0 or 1>L V U. r>0 and 2<L The total wavefunction is subject to the time-independent Schrödinger equation = EV (2) 2m ar2 +V where E is the energy of the quantum particle in question and m is the mass of the quantum particle. A The total wavefunction of a free particle that enters the barrier from...
For the particle-in-a-box of length a, assume that instead of a sine function, the ground state wavefunction is an upside-down parabola at the center of the box, b/2. What is the total energy of...
For the particle-in-a-box of length a, assume that instead of a sine function, the ground state wavefunction is an upside-down parabola at the center of the box, b/2. What is the total energy of the trial system and what is the wavefunction of the system. Now compare your result to the particle-in-a-box where the potential energy inside the box is zero, what is the difference of percentage of both systems?
For the particle-in-a-box of length a, assume that instead of...
4. A (one dimensional) particle in a box of length 2a (i.e., zero potential energy) is...
4. A (one dimensional) particle in a box of length 2a (i.e., zero potential energy) is represented by the wavefunction v(x) 0, otherwise a. Sketch the wavefunction. Write down the (time independent) Schrodinger equation. Show whether or not the wavefunction is a solution to the equation. b. What does it mean physically if the wavefunction of the particle is NOT a solution to the Schrodinger equation? Explain. c. Determine the normalization constant A. 5. Same system. Find the average or...
Determine the energy of the particle if the proposed wave function satisfies the Schrödinger equation for x < 0.
A fellow student proposes that a possible wave function for a free particle with mass \(m\) (one for which the potential-energy function \(U(x)\) is zero ) is$$ \psi(x)=\left\{\begin{array}{ll} e^{-k x}, & x \geq 0 \\ e^{+\kappa x}, & x<0 \end{array}\right. $$where \(\kappa\) is a positive constant. (a) Graph this proposed wave function.(b) Determine the energy of the particle if the proposed wave function satisfies the Schrödinger equation for \(x<\)0.(c) Show that the proposed wave function also satisfies the Schrödinger equation...
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,...
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...
A particle is trapped in a one-dimensional potential energy well given by: 100 x < 0...
A particle is trapped in a one-dimensional potential energy well given by: 100 x < 0 0 < x <L U(x) = L < x < 2L (20. x > 2L Consider the case when U, < E < 20., where E is the particle energy. a. Write down the solutions to the time-independent Schrödinger equation for the wavefunction in the four regions using appropriate coefficients. Define any parameters used in terms of the particles mass m, E, U., and...
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...
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) =...
(III) Quantum Tunneling Consider an electron in 1D in presence of a potential barrier of width L represented by a step function ſo I<0 or 1>L V U. r>0 and 2<L The total wavefunction is subject to the time-independent Schrödinger equation = EV (2) 2m ar2 +V where E is the energy of the quantum particle in question and m is the mass of the quantum particle. A The total wavefunction of a free particle that enters the barrier from...
For the particle-in-a-box of length a, assume that instead of a sine function, the ground state wavefunction is an upside-down parabola at the center of the box, b/2. What is the total energy of the trial system and what is the wavefunction of the system. Now compare your result to the particle-in-a-box where the potential energy inside the box is zero, what is the difference of percentage of both systems?
For the particle-in-a-box of length a, assume that instead of...
4. A (one dimensional) particle in a box of length 2a (i.e., zero potential energy) is represented by the wavefunction v(x) 0, otherwise a. Sketch the wavefunction. Write down the (time independent) Schrodinger equation. Show whether or not the wavefunction is a solution to the equation. b. What does it mean physically if the wavefunction of the particle is NOT a solution to the Schrodinger equation? Explain. c. Determine the normalization constant A. 5. Same system. Find the average or...
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,...
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...
A particle is trapped in a one-dimensional potential energy well given by: 100 x < 0 0 < x <L U(x) = L < x < 2L (20. x > 2L Consider the case when U, < E < 20., where E is the particle energy. a. Write down the solutions to the time-independent Schrödinger equation for the wavefunction in the four regions using appropriate coefficients. Define any parameters used in terms of the particles mass m, E, U., and...
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