a) If the wave function is normalized then the square of the absolute value of the coefficients should add up to 1
Hence the wave function is normalized.
b)
Note that
The wave function is not an eigenfunction of the Hamiltonian because
This cannot be written as
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)...
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?
Problem 3: Time-Independent Perturbation Theory Consider the particle in a 1D box of size L, as in Fig. 3. A perturbation of the form. V,δ ((x-2)2-a2) with a < L is applied to the unperturbed Hamiltonian of the 1D particle in a box (solutions on the equation sheet). Here V is a constant with units of energy. Remember the following propertics of the Dirac delta function m,f(x)6(x-a)dx f(a) 6(az) が(z) = = ds( dz E, or Ψ(x)-En 10 0.0 0.2...
1. (50 points) Consider the particle in a one-dimensional box (0 s x S L). Assume a term is added to the Hamiltonian of the form: πχ V(x)g sin Sketch the potential and the expected eigenfunction (small g). In the limit of small g, find the second order correction to the ground state energy 2. (50 points) For a diatomic molecule rotating in free space, the Hamiltonian may be written: 12 21 Where L is the total angular momentum operator,...
6. (20 points) Consider a free particle of mass m in a cubical box of side L with the Hamiltonian H = - -V2. We assume periodic boundary condition (a) Find the eigenfunctions (F) and its eigenvalue E (b) In the coordinate representation, find the density matrix in the canonical ensemble That is, to find (7le-BH) (c) Find the trace of the density matrix. 6. (20 points) Consider a free particle of mass m in a cubical box of side...
2. Consider an electron in a 1D potential box (V(x) = 0 for 0<x<L, V(x) = co otherwise) of length L = 1 nm. The electron is described by the wave function, c) = Jasin ( (a) Using the appropriate Hamiltonian derive an expression for the kinetic energy of the electron (5 marks) (b) Calculate the energy (in Joules) of the transition between the ground state and the 1 excited state. [3 marks]
What is the normalized form of the wavefunction x)Ax(L-x) for a one-dimension particle in a box with length L:
P7B.8 A normalized wavefunction for a particle confined between 0 and L in the x direction, and between 0 and L in the y direction (that is, to a square of side L) is Ψ= (2/L) sin(nx/L) sin(ny/L). The probability of finding the particle between x, and x, along x, and between y, and y, along y is P- Calculate the probability that the particle is: (a) between 0 and x L/2,y O and y L/2 (i.e, in the bottom...
question 7 9 10 ), where n, a, and are constant, is an eigenfunction of p. 7. (a) p. =- what is p. ? (b) sin( i ax what is the eigenvalue? (107) (9) = v ydt for a normalized wavefunction. Please find (1) for(a) v. - and (b) 42p. 4/2008 re s in sind. (hint : integrate over all space: sin Odrdodø (sin? xdx = [l-c952de, 5 xede = (203) 3 2 10. A particle of mass m is...
(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...