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 to and 1 vo(x) = mw 1/4 Th and $₁(x) = (mw) ¹/ 1/4) 2mw ħ find the expectation value of x as a function of time. Notice that it oscillates sinusoidally. What is the amplitude of the oscillation? What is its angular frequency? we糖² (d) Use the result of part (c) to determine (p) and check that Ehrenfest's theorem holds for this wavefunction.
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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
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...
Q3) A particle in the harmonic oscillator potential has the initial normalized wave function Ψ(?, 0)...
Q3) A particle in the harmonic oscillator potential has the initial normalized wave function Ψ(?, 0) = 1 /√5 [2 ?₁ (?) + ?₂ (?)] where ?1 and ?2 are the eigenfunctions of the oscillator Hamiltonian for ? = 1,2 states. a) Write down the expression for Ψ(?,?). b) Calculate the probability density ℙ(?,?) = |Ψ(?,?)| ² . Express it as a sinusoidal function of time. To simplify the result, let ? ≡ (?² ℏ)/ 2??² . c) Calculate 〈?〉...
1) Wave function for the ground state of an harmonic oscillator is given by. (x) =...
1) Wave function for the ground state of an harmonic oscillator is given by. (x) = A1/2 (a/T)1/4 e-ax /2 Evaluate the expectation value <x<> for this wave state (ove (Hint: Joo.co u² e-a u du = 2;. ue-au du = (1/2a) (Tc/a)2) pace)
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...
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)...
First four harmonic oscillator normalized wavefunctions 1/4 Y.-(4)"-** 4, = 1/4 v2y ev2 1/4 Y, =|...
First four harmonic oscillator normalized wavefunctions 1/4 Y.-(4)"-** 4, = 1/4 v2y ev2 1/4 Y, =| -1)ev¾2 1/4 - 3y)e¬v³½ y =ax 1. Consider a harmonic oscillator with a = 1. a) Prove that these eigenstates are all orthonormal b) Plot the first four eigenstates. How would doubling the mass change the eigenfunctions? c) Pick one eigenstate, and show that it is a solution to the Schrodinger Equation, that is, show that V? on (x) + w²ma? ¢n (x) =...
3. (a) Consider a 1-dim harmonic oscillator in its ground state (0) of the unperturbed Hamiltonian at t--0o. Let a pert...
3. (a) Consider a 1-dim harmonic oscillator in its ground state (0) of the unperturbed Hamiltonian at t--0o. Let a perturbation Hi(t)--eEXe t2 (e, E and rare constants) be applied between - and too. What is the probability that the oscillator will be in the state n) (of the unperturbed oscillator) as t-> oo?(15%) (b) The bottom of an infinite well is changed to have the shape V(x)-ε sin® for 0Sxa. Calculate the energy shifts for all the excited states...
1. A particle, initially (t -> 0) in the ground state of an infinite, 1D potential...
1. A particle, initially (t -> 0) in the ground state of an infinite, 1D potential box with walls at r 0 and = a, is subjected at time t = 0 to a time-dependent perturbation V (r, t) et/7, with eo a small real number a) Calculate to first order the probability of finding the particle in an excited state for t 0. Consider all final states. Are all possible transitions allowed? b) Examine the time dependence of the...
1. A particle in an infinite square well has an initial wave function Alsin sin 4 0 < x < L other...
help on all a), b), and c) please!!
1. A particle in an infinite square well has an initial wave function Alsin sin 4 0 < x < L otherwise s(x, t = 0) 0 (a) Find A so that the wavefunction is normalized. (b) Find '(z,t). (c) Find the expectation value(E) of the energy of ψ(x,t = 0). You may use the result mx n 2 0
1. A particle in an infinite square well has an initial wave...
6. a) Calculate the expectation value of x as a function of time for an electron...
6. a) Calculate the expectation value of x as a function of time for an electron in a state that is a (normalized) equal mixture of the ground state and 1st excited state of a 1D HO b) Graph x vs time for the case k = 1 eV/nm2. What is its value at t=0? What is the period of the oscillation in femtoseconds? For the one-dimensional (1D) harmonic oscillator (HO) the potential energy function has the form V(a) k2/2,...
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...
1) Wave function for the ground state of an harmonic oscillator is given by. (x) = A1/2 (a/T)1/4 e-ax /2 Evaluate the expectation value <x<> for this wave state (ove (Hint: Joo.co u² e-a u du = 2;. ue-au du = (1/2a) (Tc/a)2) pace)
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...
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)...
First four harmonic oscillator normalized wavefunctions 1/4 Y.-(4)"-** 4, = 1/4 v2y ev2 1/4 Y, =| -1)ev¾2 1/4 - 3y)e¬v³½ y =ax 1. Consider a harmonic oscillator with a = 1. a) Prove that these eigenstates are all orthonormal b) Plot the first four eigenstates. How would doubling the mass change the eigenfunctions? c) Pick one eigenstate, and show that it is a solution to the Schrodinger Equation, that is, show that V? on (x) + w²ma? ¢n (x) =...
3. (a) Consider a 1-dim harmonic oscillator in its ground state (0) of the unperturbed Hamiltonian at t--0o. Let a perturbation Hi(t)--eEXe t2 (e, E and rare constants) be applied between - and too. What is the probability that the oscillator will be in the state n) (of the unperturbed oscillator) as t-> oo?(15%) (b) The bottom of an infinite well is changed to have the shape V(x)-ε sin® for 0Sxa. Calculate the energy shifts for all the excited states...
1. A particle, initially (t -> 0) in the ground state of an infinite, 1D potential box with walls at r 0 and = a, is subjected at time t = 0 to a time-dependent perturbation V (r, t) et/7, with eo a small real number a) Calculate to first order the probability of finding the particle in an excited state for t 0. Consider all final states. Are all possible transitions allowed? b) Examine the time dependence of the...
help on all a), b), and c) please!!
1. A particle in an infinite square well has an initial wave function Alsin sin 4 0 < x < L otherwise s(x, t = 0) 0 (a) Find A so that the wavefunction is normalized. (b) Find '(z,t). (c) Find the expectation value(E) of the energy of ψ(x,t = 0). You may use the result mx n 2 0
1. A particle in an infinite square well has an initial wave...
6. a) Calculate the expectation value of x as a function of time for an electron in a state that is a (normalized) equal mixture of the ground state and 1st excited state of a 1D HO b) Graph x vs time for the case k = 1 eV/nm2. What is its value at t=0? What is the period of the oscillation in femtoseconds? For the one-dimensional (1D) harmonic oscillator (HO) the potential energy function has the form V(a) k2/2,...
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