7.1 Application of the Schrödinger Equation to the Hydrogen Atom

7.2 Solution of the Schrödinger Equation for Hydrogen

7.3 Quantum Numbers

7.4 Magnetic Effects on Atomic Spectra – Normal Zeeman Effect

7.5 Intrinsic Spin

7.6 Energy Levels and Electron Probabilities

CHAPTER 7The Hydrogen Atom

By recognizing that the chemical atom is composed of single separable electric quanta, humanity has taken a great step forward in the investigation of the natural world.

- Johannes Stark

The Dutch physicist Pieter Zeeman showed the spectral lines emitted by atoms in a magnetic field split into multiple energy levels. It is called the Zeeman effect. If line is split into three lines ➝ Normal Zeeman effect If line splits into more lines ➝ Anomalous Zeeman effect (Chapter 8)

Normal Zeeman effect (A spectral line is split into three lines): Consider the atom to behave like a small magnet. The current loop has a magnetic moment μ = IA and the period T = 2πr / v. Think of an electron as an orbiting circular current loop of I = dq / dt

7.4: Magnetic Effects on Atomic Spectra The Normal Zeeman Effect

: Magnetic moment of Hydrogen atom

With no magnetic field to align them, point in random directions.

However in an external field the dipole will experience a torque which tends to align it in the field.

　 for any orientation the moment hasa potential energy

The Normal Zeeman Effect

Bohr magneton

B

The Normal Zeeman Effect

Depending on the orientation of the magnetic moment (value of mℓ), the potential of the orbiting electron in a magnetic field is:

Therefore each energy level (ℓ) of the orbiting electron which are (2 ℓ +1) degenerate will be split by the applied magnetic field.When a magnetic field is applied, the 2p level of atomic hydrogen is split into three different energy states with energy difference of ∆E = μBB ∆mℓ.

mℓ Energy1 E0 + μBB

0 E0

−1 E0 − μBB

The Normal Zeeman Effect

A transition from 2p to 1s

3 different spectral lines due to photons emitted with slightly different wavelengths from field split levels.

2p

1s

Example 7.7 Determine the value of the Bohr magneton, and use this value to

determine the energy difference between the mℓ = 0 and +1 components of the 2p state of atomic hydrogen placed in a 2 Tesla magnetic field.

22 1 / 2 13.6 / 4 3.4E E eV eV

Relatively speaking E is small, however easily observed optically.

In 1920’s experiments were performed to directly measure the space quantization of atoms.

In 1922 Stern and Gerlach reported results of an experiment that clearly showed evidence for space quantization.

Experiment was based on the idea that an inhomogeneous magnetic field will exert a net force (in addition to a torque) on magnetic dipole.

Stern Gerlach Experiment

Stern Gerlach Experiment

Inhomogeneous magnetic field

If it moves through a homogeneous magnetic field, the forces exerted on opposite ends of the dipole cancel each other out and the trajectory of the particle is unaffected.

An atomic beam of particles in the ℓ = 1 state pass through a magnetic field along the z direction.

The mℓ = +1 state will be deflected down, the mℓ = −1 state up, and the mℓ = 0 state will be undeflected.

If the space quantization were due to the magnetic quantum number mℓ, mℓ states is always odd (2ℓ + 1) and should have produced an odd number of lines. One single line (ℓ = 0), 3 lines (ℓ = 1), 5 lines (ℓ = 3), …

But, they observed only two lines! The atoms are deflected either up or down! Why?

Stern-Gerlach Experiment

7.5: Intrinsic Spin Clearly there was a problem with number of lines observed in the Stern

Gerlach experiment. In 1925, Samuel Goudsmit and George Uhlenbeck in Holland proposed that

the electron must have an intrinsic angular momentum and therefore a magnetic moment.

In order to explain experimental data, Goudsmit and Uhlenbeck proposed that the electron must have an intrinsic spin quantum number s = 1/2.

Paul Ehrenfest showed that the surface of the spinning electron should be moving faster than the speed of light!

Therefore this intrinsic spin angular momentum must be a purely quantum result.

( 1) S s s

3( 1) 2

S s s

: spin angular momentum

Intrinsic Spin (fourth quantum number: ms) The spinning electron reacts similarly to the orbiting electron in a

magnetic field. Spin will have quantities analogous to L, Lz, ℓ, and mℓ. like L, the electron’s spin can never be spinning with its magnetic

moment μs exactly along the z axis. The z – component of the spin angular momentum:

The magnetic spin quantum number ms has only two values, since (2s+1) = 2 depending on the value of ms , the spin will be either “up” or “down”.

Each electronic state in the atom now described by four quantum numbers (n, ℓ, mℓ, ms)

ssz mL

2121

sm

Intrinsic Spin The magnetic moment due to intrinsic spin is

. The coefficient of is −2μB

We can define the gyromagnetic ratio (for ℓ or s). Where gℓ = 1 and gs = 2, then

The z component of . We are now in a position to understand the results of the Stern Gerlach

experiment that produced only two distinct lines. If electrons are in an ℓ = 0 state no splitting due to mℓ.

But, there is space quantizationdue to the intrinsic spin, ms.

2Bem

, or 2 B

s se S Sm

, or

2Be L L

m

that of / is BL

, , ,Bs sg L S

2 , B B B Bs sg S S g L L

Intrinsic Spin So same argument that was used previously for the orbiting electron

can now be applied the magnetic moment due to the intrinsic spin. In a field the potential energy becomes:

Note: twice difference in ms levels than the difference in mℓ levels.　

Special Topic: 21 cm line transition (page 260) Both proton and electron have intrinsic spin lowest energy state is with them oppositely aligned. ∆E between opposite and parallel is 5.9 × 10-6 eV. corresponds to a photon λ of 21 cm (radio wave) widely used in Astronomy.

2B s s Be e eV B S B m B B Bm m m

2 2B Be eV B L B m B m Bm m

, 2B s BV B

,B BV B

7.6: Energy Levels and Electron Probabilities

For hydrogen, the energy level depends on the principle quantum number n. these energies are predicted with great

accuracy by the Bohr model. In a magnetic field the degeneracy is

removed.

In many-electron atoms the degeneracy is also removed either because of internal B fields or because the average potential of an electron due to nucleus plus electrons is non-Coulombic. generally smaller ℓ states tends to lie at

lower energy for a given n. For example, sodium

E(4s) < E(4p) < E(4d) < E(4f).

(hydrogen atom)

Selection rules for radiative transition In ground state an atom cannot emit radiation.

It can absorb electromagnetic radiation, or gain energy through inelastic bombardment by particles.

Can use the wave functions to calculate transition probabilities for electrons changing from one state to another.

Allowed transitions: (for photon) Electrons can absorb or emit photons to

change states when ∆ℓ = ±1. Change in ℓ implies a ∆LZ of ±ħ. Conservation of angular momentum means

photon takes up this angular momentum.

Forbidden transitions:(for photon)　 Other transitions possible but occur with much

smaller probabilities when ∆ℓ ≠ ±1.

10, 1

n anything

m

Selection rule

(참고) Radiative transitions and Selection rules- Concepts of Modern Physics, Beiser, Chapter 6.8 -

What happens when an electron goes from one state to another?

Energy level En에서 Em으로떨어질때의

복사선의 진동수 ()?

전자가 한방향 (x-축 ) 으로 움직이는 system 을고려하면 ,양자수 n인 (에너지 En인상태) 전자의 시간에의존하는파동함수 : Ψn

Ψn = (시간에무관한파동함수 n ) x (주파수 νn = En/h 인시간 의존함수)

이러한 전자의 위치에대한기대치 <x> 는

hEE nm

thiEnn

ne )/(

hEE nm

dxexdxxx tiEiEnnnn

nn //

dxx nn

Em

En

h

따라서, En 에너지상태에있는전자의 <x> 는 n과 n* 위치만의함수로 정의되기 때문에,

<x> 값은시간에 따라변하지 않는다.전자는진동하지않으므로어떠한복사(radiation)도일어나지않는다양자역학은특정한양자상태에머물러있는원자는복사하지않는다는것을 예측하여주고있고

관측결과와일치한다.

Next, consider an electron that shifts from one energy state to another. (다른 particle 과충돌혹은 radiation 을 흡수/방출하여 )

What is the frequency of radiation?

n 상태와m 상태모두있을수있는전자의파동함수 Ψ.

a*a probability that the electron is in state nb*b probability that the electron is in state m

n nx x dx

Em

En

h

n na b

(참고)

전자가위상태중어느곳에있든지복사 (radiation)을발생하지않음 .그러나, m 에서 n 으로천이하는과정 (a 와 b 어느것도 0이아님) 에서는전자기파가발생된다.

중첩된파동함수의기대값 <x>는

*n m n mx x dx x a b a b dx

dxbbaabax mmmnnmnn

*2****2

a*a + b*b = a2 + b2 = 1 a= 1 , b = 0 e in the ground state (En)a= 0 , b = 1 e in the excited state (Em)

Em

En

h( / )niE h t

n ne

n na b ( / )miE h tm me

(참고)

/ /2 * * * m niE t iE tn n m nx a x dx b a x e e dx

/ /* * 2 *n miE t iE tn m m ma b x e e dx b x dx

sincos iei

sincos ie i

dxbaabxtEEmnnm

nm

****cos

dxbaabxtEEi mnnmnm

****sin

(참고)

이결과의 Real part 는시간에따라다음과같이변한다.

따라서전자는삼각함수모양으로진동을하게되며,그진동수는

* 전자가 n 상태나m 상태에머물러있을때전자의위치에대한기대값은상수이다.진동하지않고복사도안일어난다

* 전자가두상태사이에서전이할때, 전자는진동수 로진동한다.이러한전자는 electric dipole 같이동일한진동수 ( )를갖는

electromagnetic wave 를복사한다.

tth

EEtEE nmnm 2cos2coscos

hEE nm

Em

En

h

(참고)

Some transitions are more likely to occur than others Selection rule

진동수 를 알기위해서는시간의함수로서 a, b 의확률이나파동함수 n과 m를알필요는없다.그러나 , 어떤전이가 일어날 확률을계산하기위해서는이값들이필요하다.

여기상태에있는원자가복사하기위해필요한일반적인조건은적분

이어야하는데, 그것은 복사강도가 적분값에 비례하기때문이다.

허용된전이 (allowed transition) ← 적분값 ≠0 인경우금지된전이(forbidden transition) → 적분값 =0 인경우

0*

dxx mn

(참고)

수소원자의경우복사전이에관여하는처음상태와나중상태를규정하기위해 three quantum numbers 가필요하다.

Initial state : n’, l’, ml’Final state : n, l, ml

수소원자의 n,l,ml을알고있으므로 u=x, u=y, u=z 인경우위의값을구할수있다.

그결과 l 이 +1과 –1 만큼변하고ml이바뀌지않거나 +1 또는 –1 만큼바뀌는전위만일어난다.

주양자수 n의변화는제한받지않는다.

0',','

*,, ll

mlnmlnu

선택규칙 l = +- 1 에의해허용되는전이를보여주는수소원자의에너지준위그림 .

Allowed transition

Selection rules

l = ±1ml = 0, ±1

원자가복사를일으키기위해서 l = 1 이되어야한다는 selection rule 은 photon 이처음과나중의각운동량차이에해당하는각운동량 (angular momentum) h 를갖고방출된다는것이다.

각운동량 h 를가지고있는광자란 좌원편광혹은우원편광된전자기파라는의미임.

(참고)

(참고) Normal Zeeman effect and Selection rules- Concepts of Modern Pgysics, Chapter 6.10, Beiser-

자기장 내에서는 특정 원자 상태의 에너지는 n 값 뿐만 아니라 ml 의 값에도 관계한다.

복사전이의경우, ml = 0, 1 로제한되기때문에서로다른 l 를가진갖는 두상태에서생기는 spectrum 선은3개의진동수만갖게된다.

진동수가 0인 spectrum 선이다음과같은진동수를갖는 3개의성분으로분리됨.

Bm

ehB

Bm

ehB

B

B

4

4

003

02

001

Normal Zeeman effect

Probability Distribution Functions Unlike the planetary model of Bohr, in Schrödinger wave picture

the “position” of the electron in each state, defined by 4 quantum numbers (n, ℓ, mℓ, ms), is spread over space and is not well defined.

We must use the wave functions to calculate the probability distributions of the electrons The probability of finding the electron in a

differential volume element d is:

in spherical polar coordinates:

ϕ dependence: Ig(ϕ)I2 contains factor (eimℓϕ)* eimℓϕ = 1 thus no ϕ dependence in

probability thus wave functions are symmetric about z-axis.

Probability Distribution Functions If we are only interested in the radial probability distributions

of the electron, and, if the functions f(ϴ) and g(ϕ) have been properly normalized

then integration over all values of ϴ, and ϕ yields:

Therefore,

Note: the radial probability density P(r) = r2|R(r)|2 depends only on n and l.

: Radial Probability

Observations:

< r > depends on n or energy, and is independent of ℓ.

< r > ∝ n2

(same as Bohr).

most probable radii for 1s and 2p agree with Bohr result (a0 and 4a0respectively). true for wave functions with maximum ℓ

Probability Distribution Functions( )nR r

22( )n nP r r R

R(r) and P(r) for the lowest-lying states of the hydrogen atom

Probability Distribution Functions( )nR r

22( )n nP r r R

Example 7.11Find the most probable radius for the electron in a hydrogen atom in the 1s state.

To find the maximum we set the derivative of the radial probability equal to zero.

Example 7.12: Calculate the average orbital radius of 1s. < r > = (3/2) a0

Probability Distribution Functions 22( )n nP r r R

Example 7.13What is the probability of an electron in the 1s state of the hydrogen atom being at a radius greater than the Bohr radius?

Probability Distribution Functions 22( )n nP r r R

The electron “cloud” in the hydrogen atom

Probability Distribution Functions

( , , ) , ,n m r n m