![Question 1 The Radial Equation for an electron in hydrogen atom can be given by [Notes, Equation SR10] (n +D! j! (21 + 1 +j)(](http://img.homeworklib.com/questions/3cd7fe40-519a-11eb-bd04-a5b01355baa3.png?x-oss-process=image/resize,w_560)
Homework Answers
The function is given in terms of 'a' which itself is Bohr radius as mentioned, so I did not use 'a0' there.
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Question 1 The Radial Equation for an electron in hydrogen atom can be given by [Notes,...
(1) The ground-state wave function for the electron in a hydrogen is given by ls 0...
(1) The ground-state wave function for the electron in a hydrogen is given by ls 0 Where r is the radial coordinate of the electron and a0 is the Bohr radius (a) Show that the wave function as given is normalized (b) Find the probability of locating the electron between rF a0/2 and r2-3ao/2. Note that the following integral may be useful n! 0 dr =-e re /a roa r a Ta
Radial Functions 1. The radial Schrödinger equation (in atomic units) 1 (d2R 2 dR (1)1 HR =- has ...
Radial Functions 1. The radial Schrödinger equation (in atomic units) 1 (d2R 2 dR (1)1 HR =- has the normalized analytic solutions (radial functions): 2+1 List these functions in 'Tab!eForm, up to n=2 Plot the corresponding radial distributions (r2 R2), also in 'TableForm'. Calculate the maximum probability radius for the 1s state (n=1, 0). This involves finding the location of the maximum of r2R0. d.) Show that the Is Radial function is orthogonal to that of the 2s. DO (R101...
1) (60 points) The ground state of the hydrogen atom: In three dimensions, the radial part...
1) (60 points) The ground state of the hydrogen atom: In three dimensions, the radial part of the Schrodinger equation appropriate for the ground state of the hydrogen atom is given by: ke2 -ħ2 d2 (rR) = E(rR) 2me dr2 where R(r) is a function of r. Here, since we have no angular momentum in the ground state the angular-momentum quantum number /=0. (a) Show that the function R(r) = Ae-Br satisfies the radial Schrodinger equation, and determine the values...
Problem 10 (Problem 2.24 in textbook) The wavefunction for the electron in a hydrogen atom in its...
Problem 10 (Problem 2.24 in textbook) The wavefunction for the electron in a hydrogen atom in its ground state (the 1s state for which n 0, l-0, and m-0) is spherically symmetric as shown in Fig. 2.14. For this state the wavefunction is real and is given by exp-r/ao h2Eo 5.29 x 10-11 m. This quantity is the radius of the first Bohr orbit for hydrogen (see next chapter). Because of the spherical symmetry of ịpo, dV in Eq. (2.56)...
The average value of the radius r for a radial function Rn,l(r) of a hydrogen-like atom:...
The average value of the radius r for a radial function Rn,l(r)
of a hydrogen-like atom:
The most probable value of the radius rmp is located
where:
Calculate < r > and rmp for a hydrogen-like
atom with charge Z in the 1s and 2s states. You will find the
necessary integral and Rn,l(r) formulas on the equation
sheet. You may use numerical software or your graphing calculator
to find the roots of the cubic polynomial that you should get...
2. The hydrogen atom [8 marks] The time-independent Schrödinger equation for the hydrogen atom in...
2. The hydrogen atom [8 marks] The time-independent Schrödinger equation for the hydrogen atom in the spherical coordinate representation is where ao-top- 0.5298 10-10rn is the Bohr radius, and μ is the electon-proton reduced mass. Here, the square of the angular momentum operator L2 in the spherical coordinate representation is given by: 2 (2.2) sin θー sin θ 00 The form of the Schrödinger equation means that all energy eigenstates separate into radial and angular motion, and we can write...
Calculate the expectation value <r> of an electron in the state of n=1 and 1-0 of the hydrogen atom. r is the pos...
Calculate the expectation value <r> of an electron in the state of n=1 and 1-0 of the hydrogen atom. r is the position from the nucleus. Use the wave functions appropriately in Table 6-1 of the textbook. You can use the integration of x" exp(-ax) dx= a (n>-1, a>0). an+1
Calculate the expectation value of an electron in the state of n=1 and 1-0 of the hydrogen atom. r is the position from the nucleus. Use the wave functions appropriately...
Consider a particle of mass m in an infinite spherical potential well of radius a For write down the energies and corresponding eigen functions ψ--(r,0.9). (3 pt) a) ne that at t-o the wave fu...
Consider a particle of mass m in an infinite spherical potential well of radius a For write down the energies and corresponding eigen functions ψ--(r,0.9). (3 pt) a) ne that at t-o the wave function is given by o)-A. Find the normalization constant A function in this basis. Solve for the coeffici You may find useful the integrals in the front of the (6 pt) d) Now consider the finite potential spherical well with V(r)- ing only the radial part...
H-like atom: Bohr's model part 1 You carry out a theoretical work on absorption of alpha...
H-like atom: Bohr's model part 1 You carry out a theoretical work on absorption of alpha rays, passing on to a study of the structure of atoms on the basis of Rutherford's discovery of the atomic nucleus. By introducing conceptions borrowed from Quantum Theory established by Planck, you succeeded in working out and presenting a picture of atomic structure that, with later improvements, still fitly serves as an elucidation of the physical & chemical properties of the elements. You started...
question number 5 6. Suppose that the electron in a hydrogen atom is perturbed by a...
question number 5
6. Suppose that the electron in a hydrogen atom is perturbed by a repulsive potential concentrated at the origin. Assume the potential has the form of a 3-dimensional delta function, so the perturbed Hamiltonian is pe? H +.48°(r). 2m T where A is a constant. To first order in A, find: (a) the change in the energy of the state with quantum numbers n = 1,1 = 0. Hint: 1100(r) = 2 exp(-r/ao) 3/2 V4π αο (b)...
(1) The ground-state wave function for the electron in a hydrogen is given by ls 0 Where r is the radial coordinate of the electron and a0 is the Bohr radius (a) Show that the wave function as given is normalized (b) Find the probability of locating the electron between rF a0/2 and r2-3ao/2. Note that the following integral may be useful n! 0 dr =-e re /a roa r a Ta
Radial Functions 1. The radial Schrödinger equation (in atomic units) 1 (d2R 2 dR (1)1 HR =- has the normalized analytic solutions (radial functions): 2+1 List these functions in 'Tab!eForm, up to n=2 Plot the corresponding radial distributions (r2 R2), also in 'TableForm'. Calculate the maximum probability radius for the 1s state (n=1, 0). This involves finding the location of the maximum of r2R0. d.) Show that the Is Radial function is orthogonal to that of the 2s. DO (R101...
1) (60 points) The ground state of the hydrogen atom: In three dimensions, the radial part of the Schrodinger equation appropriate for the ground state of the hydrogen atom is given by: ke2 -ħ2 d2 (rR) = E(rR) 2me dr2 where R(r) is a function of r. Here, since we have no angular momentum in the ground state the angular-momentum quantum number /=0. (a) Show that the function R(r) = Ae-Br satisfies the radial Schrodinger equation, and determine the values...
Problem 10 (Problem 2.24 in textbook) The wavefunction for the electron in a hydrogen atom in its ground state (the 1s state for which n 0, l-0, and m-0) is spherically symmetric as shown in Fig. 2.14. For this state the wavefunction is real and is given by exp-r/ao h2Eo 5.29 x 10-11 m. This quantity is the radius of the first Bohr orbit for hydrogen (see next chapter). Because of the spherical symmetry of ịpo, dV in Eq. (2.56)...
The average value of the radius r for a radial function Rn,l(r)
of a hydrogen-like atom:
The most probable value of the radius rmp is located
where:
Calculate < r > and rmp for a hydrogen-like
atom with charge Z in the 1s and 2s states. You will find the
necessary integral and Rn,l(r) formulas on the equation
sheet. You may use numerical software or your graphing calculator
to find the roots of the cubic polynomial that you should get...
2. The hydrogen atom [8 marks] The time-independent Schrödinger equation for the hydrogen atom in the spherical coordinate representation is where ao-top- 0.5298 10-10rn is the Bohr radius, and μ is the electon-proton reduced mass. Here, the square of the angular momentum operator L2 in the spherical coordinate representation is given by: 2 (2.2) sin θー sin θ 00 The form of the Schrödinger equation means that all energy eigenstates separate into radial and angular motion, and we can write...
Calculate the expectation value <r> of an electron in the state of n=1 and 1-0 of the hydrogen atom. r is the position from the nucleus. Use the wave functions appropriately in Table 6-1 of the textbook. You can use the integration of x" exp(-ax) dx= a (n>-1, a>0). an+1
Calculate the expectation value of an electron in the state of n=1 and 1-0 of the hydrogen atom. r is the position from the nucleus. Use the wave functions appropriately...
Consider a particle of mass m in an infinite spherical potential well of radius a For write down the energies and corresponding eigen functions ψ--(r,0.9). (3 pt) a) ne that at t-o the wave function is given by o)-A. Find the normalization constant A function in this basis. Solve for the coeffici You may find useful the integrals in the front of the (6 pt) d) Now consider the finite potential spherical well with V(r)- ing only the radial part...
question number 5
6. Suppose that the electron in a hydrogen atom is perturbed by a repulsive potential concentrated at the origin. Assume the potential has the form of a 3-dimensional delta function, so the perturbed Hamiltonian is pe? H +.48°(r). 2m T where A is a constant. To first order in A, find: (a) the change in the energy of the state with quantum numbers n = 1,1 = 0. Hint: 1100(r) = 2 exp(-r/ao) 3/2 V4π αο (b)...
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