Chapter 2: One Electron Atoms

Question 1

What is a one electron atom?

Answer 1

• Atom or ion with one electron bound to the nucleus or ion core with charge +Z_coree
• where e is the charge of an electron.

Question 2

The two particles in these systems interact with via the Coulomb interaction, for which the potential energy is:

Answer 2

V(r)= -Z_coree²/4π€₀r

where €₀ is the permittivity of free space and r is the inter-particle seperation.

Question 3

If the ion core is considered to be infintely heavy, the Hamiltonian governing the motion of the electron is:

Answer 3

H^=-(hbar²/2m_e)►²-(Z_coree²/4Pi €₀r)

where hbar=h/2pi, h=Planck’s constant, and m_e is the mass of an electron.

Question 4

The energy eigenvalues associated with the Hamiltonian system can be obtained using__. Which is:

Answer 4

• Schrodingers equation
• Hhat psi=E psi
• where psi is the wavefronts of the electronic states

Question 5

Solving Schrodingers equation leads to a set of energy eigenvalues of the form:

Answer 5

E_n=-(m_eZ_core²e⁴/32Pi²€₀²hbar²n²

where n=1,2,3… is the principal quantum number

Question 6

Why is there a discrepency between the calculated value of E_n and the experimentally calculated value of E_n

Answer 6

We assume that the ion core (proton) can be treated as infinitely heavy

Question 7

Photoemission eq:

Answer 7

hv=E_i-E_f

Question 8

Photoabsorption eq:

Answer 8

hv=E_f-E_i

Question 9

The Rydberg constant obtained for an electron bound to an infinitely heavy ion core is

Answer 9

R=m_eZ_core²e⁴/64Pi³ €₀² hbar³ c

Question 10

When the Z_core=H, the value of the Rydberg constant is:

Answer 10

R_inf=m_ee⁴/64Pi³€₀³ hbar³ c

Question 11

How do we correct the differences for Rydberg constant

Answer 11

To do this we consider the reduced mass u_m of the two body system. This is the effective mass of the system in the frame of reference associated with the centre of mass.

Question 12

Equation for reduced mass:

Answer 12

u_m=m_em_p/m_e+m_p

Question 13

General Equation for the Rydberg constant (with reduced mass):

Answer 13

R_m=R_inf (u_m/m_e ) Z_core²

Question 14

atomic unit of mass:

atomic unit of charge:

atomic unit of length:

atomic unit of energy:

Answer 14

m_e mass of electron

e charge of electron

a₀ Bohr radius

2hcR_m

Question 15

Spherical Polar coordinates x,y,z :

Answer 15

x=rcosthetasinphi

y=rcosphisintheta

z=rcosphi

Question 16

Wavefunction equation and how is it seperable:

Answer 16

Psi(r,theta,phi)=R(r)Y(theta,phi)

Question 17

Classical orbital momentum:

Answer 17

L=r x p

Question 18

Effect the correspondence principle has on r and p

Answer 18

r=rhat

p=-ihbar►

Question 19

Movement of angular momentum:

Answer 19

Processes about the z-axis.

Question 20

What are the simultaneous eigenvalue equations

Answer 20

The spherical harmonic function Y_l,m(theta,phi)

Question 21

What do l and m stand for

Answer 21

l=orbital angular momentum quantum number

m=azimuthal quantum number

Question 22

Hydrogen Symbol, composition and Zcore

Answer 22

H, p⁺-e^-,+1e

Question 23

Deuterium Symbol, composition and Zcore

Answer 23

D,np⁺-e^-, +1e

Question 24

Positronium Symbol, composition and Zcore

Answer 24

Ps, e⁺-e^-, +1e

Question 25

Describe parity

Answer 25

if (-1)^l=+1, even parity

if (-1)^l=-1, odd parity

Question 26

What does l stajnd for and what are the levels

Answer 26

Orbital angular momentum quantum number

S P D F G H

0 1 2 3 4 5

Question 27

How do you get P_n.l(r) and what is it

Answer 27

This function gives the radial probability distribution of the electron. This is the probability of finding the electron at a distance r from the ion core.

Square the modulus of the radial wavefunction and multiply by r²

Question 28

How to get the average radial position

Answer 28

Integrate between infinity and 0, for r multilpied by P_n,l(r). (radial probability distribution)

Question 29

Why does the Rydberg constant for the H atom, RH, differ from R∞?

Answer 29

R∞ is the Rydberg constant for an electron bound to a positive point particle of infinite mass. The centre-of-mass of this system is fixed at the position of this positive core. RH is the Rydberg constant corrected for the reduced mass of the electron-proton system in which the center of mass is located outside the proton.

Question 30

What functions are denoted Y_l,m(θ, φ)?

• In the function Y`,m(θ, φ) what do the variables l and m represent? - For a particular value of n what are the possible values of l and m?

Answer 30

The set of functions denoted Y_l,m(θ, φ) are the spherical harmonic functions. ` represents the orbital angular momentum quantum number m represents the azimuthal quantum number – the projection of the orbital angular momentum vector onto the z axis. ` = 0, 1, 2, 3, . . .(n − 1) m = m = 0, ±1, ±2, ±3, . . . ±

Question 31

What quantum numbers does the radial wavefunction of the hydrogen atom depend on?

Answer 31

Radial wavefunctions depend only on n and l

Question 32

Why does the probability of finding the electron close to the nucleus decrease as the value of l increases?

Answer 32

As the orbital angular momentum of the electron (the value of ) increases, the centrifugal barrier pushes it away from the nucleus. The corresponding centrifugal forces are therefore what cause the probability of finding the electron close to the nucleus to decrease as increases.

Question 33

What is meant by ml-degeneracy when referring to the energy-level structure of atoms, and why are states with different values of ml degenerate in a one-electron atom?

Answer 33

ml-degeneracy refers to the energy degeneracy (equal energy) of atomic energy levels with the same value n and ` but different values of ml . ml-degeneracy is a consequence of a central potential, such as the Coulomb potential in a one-electron atom that depends only on the radial distance r = |~r| of the electron from the nucleus and has no angular dependence.

Question 34

For a transition to occur between two energy levels in a one electron atom what must the difference in l between the two levels be?

Answer 34

The selection rules for single-photon electric-dipole transitions in one-electron atoms require that for a transition to occur the difference in l between the initial and final states is ∆l = ±1.

Question 35

Rydberg formula for energy eigenstates (wave numbers)

Answer 35

E_n/hc=-R_m/n²

Question 36

how to change a₀ to am

Answer 36

a_m=a₀ me/u_m 1/Z_core