Homework Answers
When a wavefunction is inverted and if it has the same sign at the two points, the wavefunction is said to be gerade(g) ,otherwise it is said to be ungerade(u). The first four levels of a 1D partical-in-a-box wavefuction and first four levels of harmonic oscillator are given below
1D partical-in-a-box wavefuction
Harmonic oscillator
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Give the parities (g,u) of the a) first four levels of a 1D particle-in-a-box wavefunction, and...
A particle in a 1D box is described by the following normalized wavefunction: 40(x) = (*)"'*...
A particle in a 1D box is described by the following normalized wavefunction: 40(x) = (*)"'* sin() Determine the probability that the particle will be found in the region 3
8. Consider one electron in a 1D box of side L. Its wavefunction is given by...
8. Consider one electron in a 1D box of side L. Its wavefunction is given by V3 V3 2V3i where ф1(x), фг(x), and фз(x) are the first 3 eigenfunctions of the Hamiltonian, A, of a particle in a 1D box, h2 d2 a) Is Ψ(x) normalized? If it is not normalized it, normalize it! b) Is ų (x) an eigenfunction of A? If it is an eigenfunction, what is the eigenvalue?
8. Consider one electron in a 1D box of side L. Its wavefunction is given by...
8. Consider one electron in a 1D box of side L. Its wavefunction is given by из where ф1(x), фг(x), and фз(x) are the first 3 eigenfunctions of the Hamiltonian, H, of a particle in a 1D box, 2m dx2 a) Is Ψ(x) normalized? If it is not normalized it, normalize it! b) Is Ψ(x) an eigenfunction of A? If it is an eigenfunction, what is the 9. A linear polyene contains 8 -electrons, and absorbs light with412 nm. b)...
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...
For a particle in a 1D box with a box length of 1.0 nm, a) Calculate...
For a particle in a 1D box with a box length of 1.0 nm, a) Calculate the energy separation between states n-1 and n -2 in eV. b) Calculate the energy separation between states n 8 and n 9 in eV. c) Describe how the energy separation between adjacent energy levels (n and n+1) 4. changes as n increases.
Problem #1 The explicit wavefunction for a particle in the n-1 state of the quantum harmonic...
Problem #1 The explicit wavefunction for a particle in the n-1 state of the quantum harmonic oscillator is p1(x)- Axe-bx2 where mo 2h and ?1/4 (Note: In last week's homework there was an "h" where there should have been ?. This has been corrected in this week's assignment.) (a) By applying the lowering operator to ),obtain an explicit form for o(x) (i.e. the n-0 wavefunction) (b) By applying the raising operator to x), obtain an explicit form for p2(x) (i.e....
3. A particle is in a 1D box (infinite potential well) of dimension, a, situated symmetrically ab...
3. A particle is in a 1D box (infinite potential well) of dimension, a, situated symmetrically about the origin of the x-axis. A measurement of energy is made and the particle is found to have the ground state energy: 2ma The walls of the box are expanded instantaneously, doubling the well width symmetrically about the origin, leaving the particle in the same state. a) Sketch the initial potential well making it symmetric about x - 0 (note this is different...
Please finish this question with step-by-step details, thx! Consider one electron in a 1D box of...
Please finish this question with step-by-step details, thx!
Consider one electron in a 1D box of side L. Its wavefunction is given by V3 /3 A. where φ1(x),P2(x), and φ3(x) are the first 3 eigenfunctions of the Hamiltonian, H, of a particle in a 1D box, h2 d2 2mdx2 Az- a) ls (x) normalized? If it is not normalized it, normalize it! b) is Ψ(x) an eigenfunction of H? If it is an eigenfunction, what is the eigenvalue?
Q 1: For particle in a box problem, answer the following questions, a) Why n=0 is...
Q 1: For particle in a box problem, answer the following questions, a) Why n=0 is not an allowed quantum number? b) En = 0 is not allowed for particle in a box, why? c) Ground state wavefunction is orthogonal to the first excited state wavefunction, what does it mean? Q 2: An electronic system that is treated as particle in 3-D box with dimensions of 3Å x 3Å x 4Å. Calculate the wavelength corresponding to the lowest energy transition...
Particle with a speed bump Consider our old friend the 1D particle in the box, except...
Particle with a speed bump Consider our old friend the 1D particle in the box, except now with a speed bump in the box so the potential now is given by L and L < x < L 0, Vo 0 x V (x) otherwise (a) Calculate the first order correction to the ground state (n = 1) and first excited state (n = 2) energies (b) Calculate the first order correction to the ground state wave function in terms...
A particle in a 1D box is described by the following normalized wavefunction: 40(x) = (*)"'* sin() Determine the probability that the particle will be found in the region 3
8. Consider one electron in a 1D box of side L. Its wavefunction is given by V3 V3 2V3i where ф1(x), фг(x), and фз(x) are the first 3 eigenfunctions of the Hamiltonian, A, of a particle in a 1D box, h2 d2 a) Is Ψ(x) normalized? If it is not normalized it, normalize it! b) Is ų (x) an eigenfunction of A? If it is an eigenfunction, what is the eigenvalue?
8. Consider one electron in a 1D box of side L. Its wavefunction is given by из where ф1(x), фг(x), and фз(x) are the first 3 eigenfunctions of the Hamiltonian, H, of a particle in a 1D box, 2m dx2 a) Is Ψ(x) normalized? If it is not normalized it, normalize it! b) Is Ψ(x) an eigenfunction of A? If it is an eigenfunction, what is the 9. A linear polyene contains 8 -electrons, and absorbs light with412 nm. b)...
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...
For a particle in a 1D box with a box length of 1.0 nm, a) Calculate the energy separation between states n-1 and n -2 in eV. b) Calculate the energy separation between states n 8 and n 9 in eV. c) Describe how the energy separation between adjacent energy levels (n and n+1) 4. changes as n increases.
Problem #1 The explicit wavefunction for a particle in the n-1 state of the quantum harmonic oscillator is p1(x)- Axe-bx2 where mo 2h and ?1/4 (Note: In last week's homework there was an "h" where there should have been ?. This has been corrected in this week's assignment.) (a) By applying the lowering operator to ),obtain an explicit form for o(x) (i.e. the n-0 wavefunction) (b) By applying the raising operator to x), obtain an explicit form for p2(x) (i.e....
3. A particle is in a 1D box (infinite potential well) of dimension, a, situated symmetrically about the origin of the x-axis. A measurement of energy is made and the particle is found to have the ground state energy: 2ma The walls of the box are expanded instantaneously, doubling the well width symmetrically about the origin, leaving the particle in the same state. a) Sketch the initial potential well making it symmetric about x - 0 (note this is different...
Please finish this question with step-by-step details, thx!
Consider one electron in a 1D box of side L. Its wavefunction is given by V3 /3 A. where φ1(x),P2(x), and φ3(x) are the first 3 eigenfunctions of the Hamiltonian, H, of a particle in a 1D box, h2 d2 2mdx2 Az- a) ls (x) normalized? If it is not normalized it, normalize it! b) is Ψ(x) an eigenfunction of H? If it is an eigenfunction, what is the eigenvalue?
Particle with a speed bump Consider our old friend the 1D particle in the box, except now with a speed bump in the box so the potential now is given by L and L < x < L 0, Vo 0 x V (x) otherwise (a) Calculate the first order correction to the ground state (n = 1) and first excited state (n = 2) energies (b) Calculate the first order correction to the ground state wave function in terms...
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