Lecture 17: Radial Wavefunction and Orbital Levels of the Hydrogen Atom The material in this lecture covers the following in Atkins.

The structure and Spectra of Hydrogenic Atoms Section 13.1 The structure of hydrogenic atoms (b) The radial solution 13.2 Atomic orbitals and their energies (a) The energy levels (b) Ionization Energies Lecture on-line Hydrogen energy levels and radial wavefunction (PDF Format) Hydrogen energy levels and radial wavefunction (PowerPoint)

Hydrogen Levels

The wavefunction of the hydrogenic atom depends on three quantum numbers

(r, , R (r)Y ( ,nl l,mψ φ θ φ θ) )=

The principle quantum number : n = 1,2,3,4,....Angular momentum quantum number : l = 0,1,2,3,..n - 1Magnetic quantum number : ml = − − −l l l l, , , ....,1 2

E = -Z

32

2

2µ

π εe

no

4

2 2 2h

Ze

r

-e

φ

θ (r,φ,θ)

x

y

z

Re :view Quantum numbersof the hydrogenic atom

Hydrogen Levels

x

y

z

L2 = h2l(l + 1)

Lz= hml

Angular momentum quantum number : l = 0,1, 2, 3, ..n - 1is related to the length L of the angular momentum of theelectron as it moves around nucleus

Magnetic quantum number : m

related to the length of the projection of L on toan arbitrary vector (e

l

z

= − − −l l l l

is

, , , ....,

) .

1 2r

r

x

yz

r v

v L = v r × v v

Re :view Quantum numbersof the hydrogenic atom

Hydrogen Levels

With the radial part given as

R r Nn

L r enl nll

n ln( ) ( ),

/=

−ρ ρ 2

Normalization Polynomial Exponential

The wavefunction of the hydrogenic atom is given by :

(r, , R(r) Y ( ,nl l,mψ φ θ φ θ) )=

Ze

r

-e

φ

θ (r,φ,θ)

x

y

z

Re :( )

viewr

Radial wavefunction R of the hydrogenic atomnl

Hydrogen Levels

n = 1 ; l = 0

R r e1 0

3 22

,

//( ) =

− 2Zao

ρ

No r nodes. R everywhere positive1,0 ( )

R10 0( ) 0≠

Re :( )

viewr

Radial wavefunction R of the hydrogenic atomnl

ρ πε= =2 2

2Zr

a m eo e and a

4o

oh

Hydrogen Levels

R r e20

3 242

12

( ) ( )/

/=

− − 1

2 2Zao

ρ ρ R r e2 1

3 24

,

//( ) =

− 1

4 6Z

aoρ ρ

R : one node

R : zero nodes2,0

2,1

( )

( )

r

r

Only r o R with l = 0 has Rnl nl( ) ( ) ≠ 0

Re :( )

viewr

Radial wavefunction R of the hydrogenic atomnl

ρ πε= =2 2

2Zr

a m eo e and a

4o

oh

Hydrogen Levels

R r e30

3 22 66 2

19

( ) ( )/

/=

− + − 1

9 3Zao

ρ ρ ρ

R r e31

3 264

13

( ) ( )/

/=

− − 1

27 6Zao

ρ ρ ρ

R rR r

30

31

( )( )

: 2 nodes : 1 node

Only r o R with l = 0 has Rnl nl( ) ( ) ≠ 0

Re :( )

viewr

Radial wavefunction R of the hydrogenic atomnl

Hydrogen Levels

R r e3 2

3 22 6

,

//( ) =

− 1

81 30Zao

ρ ρ

No r nodes. R everywhere positive3,0 ( )

Only r o R with l = 0 has Rnl nl( ) ( ) ≠ 0

In general number of nodes n - l - 1

Re :( )

viewr

Radial wavefunction R of the hydrogenic atomnl

Orbitals of Hydrogenic Atom..PE vz. KE.The balance of kinetic and potentialenergies that accounts for thestructure of the ground state ofhydrogen (and similar atoms).

(a) The sharply curved but localizedorbital has high mean kinetic energy,but low mean potential energy

(b) the mean kinetic energy is low,but the potential energy is not veryfavourable;

; (c) the compromise of moderatekinetic energy and moderatelyfavourable potential energy.

V(r) = − 1r

Hydrogen Levels

The energy levels of the hydrogenatom showing the subshells and(in square brackets) the numbersof orbitals in each subshell. Inhydrogenic atoms, all orbitals of agiven shell have the same energy.

E = -Z

32

2

2µ

π εe

no

4

2 2 2h

he organization of orbitalsnto subshellscharacterized by I)and shellscharacterized by n).

n = 1,2,3,4,....

l = 0,1,2,3,..n - 1ml = − − −l l l l, , , ....,1 2

The

hc

nhc

eZ

o

energy is related to n by

E Z

n

2= RR

RR ==ZZZ

2

4

2 2 232

µπ ε h

Hydrogen Levels

For the hydrogen atom with Z = 1

Hhc

eH

oRR ==

µπ ε

4

2 2 232 h

Introducing the Rydberg constant

we have

RR

RR == RR RRm e

h c me

oH

H

e

4

2 2 332π ε

µ=

µZZ

Z

MM

The

= m

m

reduced mass

e

e +

E n = hc

n

RR H2

Hydrogen Levels

Special feature on measuring the ionizationpotential of the hydrogen atom from its Lymanemission spectrum

-hcRRRR n=1

-hcRRRR4

n=2

-hcRRRR9

n=3

n=4-hcRRRR

16

n=5

-hcRRRR25

Ionization

e I

energy I = hc

Energy required to remove electron from groundstate

H - - > H+

RR

+ −−

EmissionExcited

: electrons

return to ground -state

Hydrogen Levels

-hcRRRR n=1

-hcRRRR4

n=2

-hcRRRR9

n=3

n=4-hcRRRR

16

n=5

-hcRRRR25

Hydrogen Levels

-hcRRRR n=1

-hcRRRR4

n=2

-hcRRRR9

n=3

n=4-hcRRRR

16

n=5

-hcRRRR25

∆E

hchc( )2 1

4→ = − +RR

RR

∆E

hchc( )3 1

9→ = − +RR

RR

∆E

hchc( )4 1

16→ = − +RR

RR

∆E n

hc

nhc( )→ = − +1 2

RRRR

Energy changes E(n 1) due toemission to groundstate

∆ →

Hydrogen Levels

-hcRRRR n=1

-hcRRRR4

n=2

-hcRRRR9

n=3

n=4-hcRRRR

16

n=5

-hcRRRR25

∆E

hchc( )2 1

4→ = − +RR

RR

Frequency of light (n 1) due toemission to groundstate

ν →

∆E

hchc( )3 1

9→ = − +RR

RR

∆E

hchc( )4 1

16→ = − +RR

RR

∆E n

hc

nhc( )→ = − +1 2

RRRR

== h (ν 2 1→ )

== h (ν 3 1→ )

== h (ν 4 1→ )

== h (ν n → 1)

Energy of light h (n 1) due to emission to groundstate

ν →

Hydrogen Levels

-hcRRRR n=1

-hcRRRR4

n=2

-hcRRRR9

n=3

n=4-hcRRRR

16

n=5

-hcRRRR25

∆E

hchc( )2 1

4→ = − +RR

RR

∆E

hchc( )3 1

9→ = − +RR

RR

∆E

hchc( )4 1

16→ = − +RR

RR

∆E n

hc

nhc( )→ = − +1 2

RRRR

== hc (tν 2 1→ )

== hc (˜ )ν 3 1→

== hc (tν 4 1→ )

== hc (tν n → 1)

Energy of light h (n 1) due to emission to groundstate

ν →

Also (n 1) (n 1) = cor

(n 1) = c/ (n 1) = c (n 1)where1/ (n 1) = (n 1)Thush (n 1) = hc (n 1)

ν λ

ν λ ν

λ ν

ν ν

→ →

→ → →

→ →

→ →

t

t

˜

Hydrogen Levels

-hcRRRR n=1

-hcRRRR4

n=2

-hcRRRR9

n=3

n=4-hcRRRR

16

n=5

-hcRRRR25

∆E nhc

nhc

n

hc

nn

( )

˜ )

˜ )

˜

→ = − +

→

− →

→ = −

1

1

1

2

2

RRRR

==

RR++ ==

hc (

hc (

(n 1)R

n+

hc 2

ν

ν

ν

I

I

˜ )ν(2 1

4→ = − RR

++ I

hc

˜ )ν(3 1

9→ = − RR

++ I

hc

˜ )ν(4 1

16→ = − RR

++ I

hc

82 250 cm-1

97 492 cm-1

102 824 cm-1

Hydrogen Levels

˜ )ν(n

n

Ihc

→ = −1 2RR

++

˜ )ν(2 1

4→ = − RR

++ I

hc

˜ )ν(3 1

9→ = − RR

++ I

hc

˜ )ν(4 1

16→ = − RR

++ I

hc

82 250 cm-1

97 492 cm-1

102 824 cm-1

Slope = -RR

Intercept = I/hc

1. You are not expected to be able to solve the Schrödinger equation for the hydrogen - like atom

H n,l,m n,l,mˆ ( , , ) ( , , )Ψ Ψr E rnϕ θ ϕ θ=

However you should be aware that the Hamiltonian can be written in the form

where r is the distance between the hydrogen - like atom and the electron whereas is the reduced mass.

r hH

r r r rL

Ze

ro= − + + −

2 2

2 22

22 1

2 4µδδ

δδ µ πε

µ

[ ]

You should also be aware of the following commutation relations

[H,L and [H,L 2zˆ ] ˆ ]= =0 0

What you need to know about the hydrogen atom from the previous lecture

2. You are not required to memorize the exact form of the eigenfunctions.

You should recognized that they can be written as a product of a radial part R (r) andthe spherical harmonics Y ( , )

[eigenfunctions of L and L ] as

(r, R (r)Y ( ,

nl

lm2

z

n,l,m

nl lm

ϕ θ

ϕ θϕ θ

ˆ ˆ

, ))

Ψ =

What you need to know about the hydrogen atom from the previous lecture

with the corresponding energies given by

E = -Z me

32p e n

2 4

2o2 2 2h

It is important that you remember the possible quantum numbers for l and m with respect toa given n.

What you need to know about the hydrogen atom from this lecture

3.( ) ( )

You should be able to count the number of nodes in R as n - l - 1 and realize that R only is differentfrom zero at the nucleus for l = 0.

n,l n,lr r

4. You are expected to relate the l and m quantum numbers to thelength of the orbital angular momentum vector and the projection of the orbital angular momentum vector on the arbitraryz - axis, respectively.

5. Review the way in which absorption spectra, emission spectraand ionization potentials for the hydrogen atom are derived from the energy levels E and make note of the degeneracy of the energylevels E (subshell structure)

n

n