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At time t = 0 a particle in a Harmonic Oscillator potential is in the state...
A particle with mass m is in a one-dimensional simple harmonic oscillator potential. At time t = 0 it is described by the state where lo and l) are normalised energy eigenfunctions corresponding to e...
A particle with mass m is in a one-dimensional simple harmonic oscillator potential. At time t = 0 it is described by the state where lo and l) are normalised energy eigenfunctions corresponding to energies E and Ey and b and c are real constants. (a) Find b and c so that (x) is as large as possible. b) Write down the wavefunction of this particle at a time t later c)Caleulate (x) for the particle at time t (d)...
A particle in the harmonic oscillator potential starts out at t = 0 in the state...
A particle in the harmonic oscillator potential starts out at t = 0 in the state Ψ(x, 0) = A (5ψ0(x) + 12ψ1(x)) (a) [2 points] Find A. (b) [5 points] Find <x> and <p> as a function of time. (c) [3 points] Check Ehrenfest’s theorem that d<p>/dt = − <dV/dx>
A particle in the harmonic oscillator potential, V(x) - m2t2, is at time t 0 in...
A particle in the harmonic oscillator potential, V(x) - m2t2, is at time t 0 in the state ψ(x, t-0) = A3ψο(x) +4ψι (2)] where vn (z) is the nth normalized eigenfunction (a) Find A so that b is normalized. (b) Find ψ(x,t) and |ψ(x, t)12 (c) Find x (t) and p)(t). what would they be if we replaced ψ1 with V2? (hint: no difficult calculations are required) Check that Ehrenfest's theorem (B&J 3.93) holds for this wavefunction. (d) What...
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...
The most general wave function of a particle in the simple harmonic oscillator potential is: V(x,...
The most general wave function of a particle in the simple harmonic oscillator potential is: V(x, t) = (x)e-1st/ where and E, are the harmonic oscillator's stationary states and their corresponding energies. (a) Show that the expectation value of position is (hint: use the results of Problem 4): (v) = A cos (wt - ) where the real constants A and o are given by: 1 2 Ae-id-1 " Entichtin Interpret this result, comparing it with the motion of a...
Suppose a particle is in a one-dimensional harmonic oscillator potential. Suppose that a perturbation is added...
Suppose a particle is in a one-dimensional harmonic oscillator
potential. Suppose that
a perturbation is added at time t = 0 of the form . Assume that at time t = 0 the
particle
is in the ground state. Use first order perturbation theory to find
the probability that at some
time t1 > 0 the particle is in the first excited state of the
harmonic oscillator.
H' = ext.
[20 points] A particle in the simple harmonic oscillator potential with angular frequency a is initially...
[20 points] A particle in the simple harmonic oscillator potential with angular frequency a is initially in the ground state: c,y, (x) =Yo(x Att = 0 , the angular frequency of the oscillator suddenly doubles: a} → a½-2.4 The initial wave function can be written in terms of the modified potential (denoted with a tilde:~: Recall that the general form of the first few stationary states for the harmonic oscillator are given on page 56 of your text. a. What...
5. A particle in the harmonic oscillator potential has the initial wave function Psi(x, 0) = A[\psi_{0}(x) + \psi_{1}(x)] for some constant A. Here to and ₁ are the normalized ground state and the first excited state wavefunctions of the harmonic oscill
5. A particle in the harmonic oscillator potential has the initial wave function Psi(x, 0) = A[\psi_{0}(x) + \psi_{1}(x)] for some constant A. Here to and ₁ are the normalized ground state and the first excited state wavefunctions of the harmonic oscillator, respectively. (a) Normalize (r, 0). (b) Find the wavefunction (r, t) at a later time t and hence evaluate (x, t) 2. Leave your answers involving expressions in to and ₁. c) sing the following normalized expression of...
For a particle described as a harmonic oscillator, the total energy w given by E,- (n...
For a particle described as a harmonic oscillator, the total energy w given by E,- (n + hy and the potential energy is piven by VG) kw The classical turning points, to are the values of x where the total energy is equal to the potential energy. The ground state wave function of a harc oscillator is . The cost is defined by a = k/?. If we define the variable y as y = x, which of the following...
A harmonic oscillator is in a state such that a measurement of the energy would yield...
A harmonic oscillator is in a state such that a measurement of the energy would yield either Eo=(1/2ho 1/3 of the time or E1-(3/2)hw, 2/3 of the time time-dependent wave function which describes this state. (b) Find the average position of the particle (x(t)). (c) Find the expectation value of the energy. Given that E(H), is the energy associate with this state constant or it oscillates in time between Eo and E1?
A particle with mass m is in a one-dimensional simple harmonic oscillator potential. At time t = 0 it is described by the state where lo and l) are normalised energy eigenfunctions corresponding to energies E and Ey and b and c are real constants. (a) Find b and c so that (x) is as large as possible. b) Write down the wavefunction of this particle at a time t later c)Caleulate (x) for the particle at time t (d)...
A particle in the harmonic oscillator potential, V(x) - m2t2, is at time t 0 in the state ψ(x, t-0) = A3ψο(x) +4ψι (2)] where vn (z) is the nth normalized eigenfunction (a) Find A so that b is normalized. (b) Find ψ(x,t) and |ψ(x, t)12 (c) Find x (t) and p)(t). what would they be if we replaced ψ1 with V2? (hint: no difficult calculations are required) Check that Ehrenfest's theorem (B&J 3.93) holds for this wavefunction. (d) What...
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...
The most general wave function of a particle in the simple harmonic oscillator potential is: V(x, t) = (x)e-1st/ where and E, are the harmonic oscillator's stationary states and their corresponding energies. (a) Show that the expectation value of position is (hint: use the results of Problem 4): (v) = A cos (wt - ) where the real constants A and o are given by: 1 2 Ae-id-1 " Entichtin Interpret this result, comparing it with the motion of a...
Suppose a particle is in a one-dimensional harmonic oscillator
potential. Suppose that
a perturbation is added at time t = 0 of the form . Assume that at time t = 0 the
particle
is in the ground state. Use first order perturbation theory to find
the probability that at some
time t1 > 0 the particle is in the first excited state of the
harmonic oscillator.
H' = ext.
[20 points] A particle in the simple harmonic oscillator potential with angular frequency a is initially in the ground state: c,y, (x) =Yo(x Att = 0 , the angular frequency of the oscillator suddenly doubles: a} → a½-2.4 The initial wave function can be written in terms of the modified potential (denoted with a tilde:~: Recall that the general form of the first few stationary states for the harmonic oscillator are given on page 56 of your text. a. What...
For a particle described as a harmonic oscillator, the total energy w given by E,- (n + hy and the potential energy is piven by VG) kw The classical turning points, to are the values of x where the total energy is equal to the potential energy. The ground state wave function of a harc oscillator is . The cost is defined by a = k/?. If we define the variable y as y = x, which of the following...
A harmonic oscillator is in a state such that a measurement of the energy would yield either Eo=(1/2ho 1/3 of the time or E1-(3/2)hw, 2/3 of the time time-dependent wave function which describes this state. (b) Find the average position of the particle (x(t)). (c) Find the expectation value of the energy. Given that E(H), is the energy associate with this state constant or it oscillates in time between Eo and E1?
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