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P10.6 Consider a charged particle of mass m in a SHO potential but which is also...
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:...
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...
Consider a particle of mass m under the action of the one-dimensional harmonic oscillator potential. The Hamiltonian is...
Consider a particle of mass m under the action of the one-dimensional harmonic oscillator potential. The Hamiltonian is given by Knowing that the ground state of the particle at a certain instant is described by the wave function mw 1/4 _mw2 Th / calculate (for the ground state): a) The mean value of the position <x> (2 marks) b) The mean value of the position squared < x2 > (2 marks) c) the mean value of the momentum <p> (2...
Consider a particle of mass in a 10 finite potential well of height V. the domain...
Consider a particle of mass in a 10 finite potential well of height V. the domain – a < x < a. a) Show that solutions for – a < x < a take the form on (x) = A cos(knx) for odd n, and on (x) = A sin(knx) for even n. . Show a) Match the boundary conditions at x = a to prove that cos(ka) = Bk where k is the wave vector for -a < x...
achieves its closest approach A particle of mass m moving in the Kepler potential V -k/...
achieves its closest approach A particle of mass m moving in the Kepler potential V -k/ to the force center, r-ro, at 0, where r, p denote polar coordinates in the plane of motion of the particle. At φ = π/3, its distance from the force center is r = 5r0/4. Determine the eccentricity e of the orbit, the angular momentum, the energy, and the ratio of speeds v(p /3)/(p 0). Hint: If you're not completely confident in your knowledge...
1 Particle in a Box with a Bump (based on B&J 4.11) Consider a particle of...
1 Particle in a Box with a Bump (based on B&J 4.11) Consider a particle of mass m in a 1-D double well with potential given by Vo, 05\x\<b V(x) = { 0, b<\x<c 100, [x]>c . We will study the lowest energy states, for which 0 <E<V, corresponding to tunnelling between the two wells. (a) Write down the time-independent Schödinger equation in the three regions -c<x<-b, –b< <b, and b< I< c. Write down the most general wavefunction solution...
A NON stationary state A particle of mass m is in an infinite square well potential of width L, a...
A NON stationary state A particle of mass m is in an infinite square well potential of width L, as in McIntyre's section 5.4. Suppose we have an initial state vector lv(t -0) results from Mclntrye without re-deriving them, and you may use a computer for your math as long as you include your code in your solution A(3E1) 4iE2)). You may use E. (4 pts) Use a computer to plot this probability density at 4 times: t 0, t2...
Consider the 1D square potential energy well shown below. A particle of mass m is about to be tra...
Consider the 1D square potential energy well shown below. A particle of mass m is about to be trapped in it. a) (15 points) Start with an expression for this potential energy and solve the Schrödinger 2. wave equation to get expressions for(x) for this particle in each region. (10 points) Apply the necessary boundary conditions to your expressions to determine an equation that, when solved for E, gives you the allowed energy levels for bound states of this particle....
1. Consider a charged particle bound in the harmonic oscillator potential V(x) = mw x2. A...
1. Consider a charged particle bound in the harmonic oscillator potential V(x) = mw x2. A weak electric field is applied to the system such that the potential energy, U(X), now has an extra term: V(x) = -qEx. We write the full Hamiltonian as H = Ho +V(x) where Ho = Px +mw x2 V(x) = –qEx. (a) Write down the unperturbed energies, EO. (b) Find the first-order correction to E . (c) Calculate the second-order correction to E ....
Question 3: A particle is in the ground state (po) of a simple harmonic oscillator potential....
Question 3: A particle is in the ground state (po) of a simple harmonic oscillator potential. (a) Determine Φ(p,t). (b) Classically, the kinetic energy cannot exceed the total mechanical energy of the particle, so w. You measure the momentum of the particle. What is the probability that you will measure a value outside of the classically allowed range? 2 Reminders: foo e-a2+br dr=v/Te4a where a is real and positive. The error e edt and can be calculated numerically function is...
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:...
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...
Consider a particle of mass m under the action of the one-dimensional harmonic oscillator potential. The Hamiltonian is given by Knowing that the ground state of the particle at a certain instant is described by the wave function mw 1/4 _mw2 Th / calculate (for the ground state): a) The mean value of the position <x> (2 marks) b) The mean value of the position squared < x2 > (2 marks) c) the mean value of the momentum <p> (2...
Consider a particle of mass in a 10 finite potential well of height V. the domain – a < x < a. a) Show that solutions for – a < x < a take the form on (x) = A cos(knx) for odd n, and on (x) = A sin(knx) for even n. . Show a) Match the boundary conditions at x = a to prove that cos(ka) = Bk where k is the wave vector for -a < x...
achieves its closest approach A particle of mass m moving in the Kepler potential V -k/ to the force center, r-ro, at 0, where r, p denote polar coordinates in the plane of motion of the particle. At φ = π/3, its distance from the force center is r = 5r0/4. Determine the eccentricity e of the orbit, the angular momentum, the energy, and the ratio of speeds v(p /3)/(p 0). Hint: If you're not completely confident in your knowledge...
1 Particle in a Box with a Bump (based on B&J 4.11) Consider a particle of mass m in a 1-D double well with potential given by Vo, 05\x\<b V(x) = { 0, b<\x<c 100, [x]>c . We will study the lowest energy states, for which 0 <E<V, corresponding to tunnelling between the two wells. (a) Write down the time-independent Schödinger equation in the three regions -c<x<-b, –b< <b, and b< I< c. Write down the most general wavefunction solution...
A NON stationary state A particle of mass m is in an infinite square well potential of width L, as in McIntyre's section 5.4. Suppose we have an initial state vector lv(t -0) results from Mclntrye without re-deriving them, and you may use a computer for your math as long as you include your code in your solution A(3E1) 4iE2)). You may use E. (4 pts) Use a computer to plot this probability density at 4 times: t 0, t2...
Consider the 1D square potential energy well shown below. A particle of mass m is about to be trapped in it. a) (15 points) Start with an expression for this potential energy and solve the Schrödinger 2. wave equation to get expressions for(x) for this particle in each region. (10 points) Apply the necessary boundary conditions to your expressions to determine an equation that, when solved for E, gives you the allowed energy levels for bound states of this particle....
1. Consider a charged particle bound in the harmonic oscillator potential V(x) = mw x2. A weak electric field is applied to the system such that the potential energy, U(X), now has an extra term: V(x) = -qEx. We write the full Hamiltonian as H = Ho +V(x) where Ho = Px +mw x2 V(x) = –qEx. (a) Write down the unperturbed energies, EO. (b) Find the first-order correction to E . (c) Calculate the second-order correction to E ....
Question 3: A particle is in the ground state (po) of a simple harmonic oscillator potential. (a) Determine Φ(p,t). (b) Classically, the kinetic energy cannot exceed the total mechanical energy of the particle, so w. You measure the momentum of the particle. What is the probability that you will measure a value outside of the classically allowed range? 2 Reminders: foo e-a2+br dr=v/Te4a where a is real and positive. The error e edt and can be calculated numerically function is...
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