Fundamentals of atomic physics

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Fundamentals of atomic physics

• Main directions of development:

- spectroscopy a) atomic b) molecular

-"new" disciplines: - nonlinear optics - quantum optics - physics of ultracold matter - quantum computing - applications- incl. quantum metrology

• Lecture schedule:

I. The atomic structure

II. Interaction of atoms with EM radiation

III. Experimental methods - important experiments in atomic physics

•Materials: http://chaos.if.uj.edu.pl/~kuba/teaching.html

• Pass - exercises + exam.

Subject of research: atom, molecule (single - not crystal or liquid)

- structure of energ. - stationary states

- influence from external factors (fields and particles)

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1665 Isaac Newton

(splitting the light into its components)

From the start …

1814 Joseph von Fraunhoffer

(absorption

lines in the solar spectrum )

1860 Robert Bunsen & Gustav Kirchhoff

(prism spectroscope)

1 - development of measurement techniques (new

observations):

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2 - Search for an explanation of the observation.

1889 Johannes R. Rydberg

÷ø ç ö

è

æ -

= ₂ 1₂ '

1 1

n R n

l

1884 Johan Jakob Balmer

(hydrogen spectrum) H

4 lines from the Fraunhoffer spectrum;

l

= (9/5) h, (4/3) h, (25/21) h, (9/8) h, where h = 364.56 nm

®

spectral series 1 /

l

~ (1/4 - 1 / n²)

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5/ 22

1. E. Rutherford's model of the atom (~ 1911)

exp. Hans Geiger and Ernest Marsden (1909)

1871-1937 Nobel 1908 (Chemistry) particle source a

(He kernels)

q

particle detector a

Metal foil.

• dispersion: alpha particle ® the repulsive Coulomb force

• Cases of Backscatter® strong res.

"® strong fields® charge ~ point

• no recoil of foil atoms ® dissipative charges in heavy "objects"

F

~ all foil matter is concentrated in a heavy nucleus atoms = extremely small, positively charged nuclei (~ 10^-14 m << atom size ~ 10^-10 m) + light electrons

The beginning of the "modern" atomic science

M

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2. Bohr's Model (1913):

"Angular momentum quantization for allowed orbits " L = mur = nħ (ħ = h / 2p)

consequences:

Only discrete circular orbits with energy- levels E_n are allowed.

Radiationless movement.

1. When switching from an orbit with greater r (greater energy) to a lower one - emission of radiation with the frequency hν= E_n-E_n’

where E_n= -Rhc / n²

2. Correspondence rule: For large n the frequency of emission /

absorption corresponds to the frequency of the orbital movement of the electron (this does not fit for small n) → we compare n and n'= n-1 →

determining the constant R.

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2. Bohr's Model (1913):

The Energy with Coulomb potential is: 𝐸 =

^!"_#^!

−

^$%^!

&

k ^≡

_()*^'

"

Example of Virial theorem: 2 𝑇 = 𝑛 ⟨𝑉⟩ for 𝑉~𝑟

⁺

forces

^!"_& ^!

=

^$%_&_!^!

𝑚𝑣

^#

/2 =

^$%_#&^!

⇒ 𝑚𝑣𝑟 ⋅ 𝑣 ⇒ 𝑣 =

^$%_,^!

From Bohr’s postulate for stationary orbits, we already have 𝐿 = 𝑛ℏ giving 𝑣 =

^$%_+ℏ^!

and thus,

𝐸

₊

= − 𝑚𝑣

^#

2 = − 𝑚𝑘

^#

𝑒

⁽

2ℏ

^#

𝑛

^#

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2. Bohr's Model (1913):

consequences:

u_n = K.Zu₀/n u₀ = e²/ ħ

Þ E._n = - (Z²/n² K.²)E._I E._I = K.me⁴/ 2ħ²= en. ionization = 13.6 eV

r_n = n² a₀/ KZ a₀ = ħ²/ me² = 0.052 nm (0.52 Å) Bohr radius K. º 1 / (4pe₀)

Sommerfeld Extensions:

- extension to elliptical orbits, quantization l = 0, .. n-1 - relativistic mass change effect - low l orbits

→ L degeneracy removed (proper - Dirac equation only) Rydberg constant: R = K.² me⁴/ 2ħ²

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9/ 22

Ground state as a steady state

G Quantum fluctuations are essential

classically

Energy E = T_clas + V_clas

T_clas = ½ mu²= |balance of forces: forces ^!"_#^! = ^$%_#_!^! | = ½ ke²/ r₀ E = - ½ ke²/ r₀

V_clas = - ke²/ r₀

E (r₀) 0.

Classically el. falls on nucleus!!

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v with Quantum. Mech. Dr Dp ³ ħ

so that the classical orbits and angular momentum make sense you need Dp << p, Dr << r, i.e (Dr / r) (Dp / p) << 1

Bohr's postulates contradictory with existing Physics

circulating electron emits (accelerated charges radiate) and it should loose energy and fall on the nucleus.

For small n

contradiction

M

F

but Dr Dp ³ ħ Þ (Dr Dp) / rp ³ ħ / rp

mvr = pr = nħ , i.e. (Dr Dp) / rp ³ 1 / n

Þ One should not talk about localized orbits Þ (in the classical sense.)

(unless n >> 1 - Rydberg states)

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11/ 22

V = -ke²/ r most preferably when r ® 0,

According to Quantum Mechanics :

v but from the uncertainty relation. when an electron is located in an area with a radius of r₀; Dr » r₀, this Dp » ħ / r₀ (non-zero

momentum)

v when momentum is nonzero, in turn, kin. En.

is nonzero.

T. ³ T._min = (Dp)²/ 2m = ħ²/ 2mr₀²

v E = T + V

minimum E_min = T_min + V occurs for r₀ = ħ²/ me² = a₀ Þ stable atom J "Zero vibration energy"

T._min

V 0 ^and⁰ r

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Quantum mechanics on atomic energy levels

electron in Coulomb field from Z protons acc. to quantum. Mech.

H_CM= p²/ 2µ - k Ze²/ r µ º m_eM / (m_e+ M), k º 1 / (4pe₀) C / r C / r coulomb (central) potential

Dy + 2µ/ħ(E-C / r) y = 0

• spherical symmetry Þ possible factorization in radial and angular parts y(r,J,j) = R (r) Y (J,j)

n = 1, 2, ...

Shrödinger eqn.:

! "#$

Possibility to separate variables in different coordinate systems - standard - spherical

- standard - parabolic, semi-parabolic

- relationships with the selection of commuting observables

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f. radial R_nl(r)

for Coulomb’s Potential R

_nl

(r) depends on n and l, but E

_n

only on n

® degeneration:

"

n, l = 0.1, ..n-1.

States m

_l

also zdegener.

Þ

degree deg.

g = S

l (2l + 1)

= n

²

V (r) does not depend on l

-13.6 -3.4 -1.51 -0.85 0

E [eV] l = 0 1 2 3 4

n = 1 n = 2 n = 3 n = 4

G

n =¥

Accidental degeneracy

(only for coulomb - hydrogen!)

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Energ levels "one-electron" atoms

)

2

²

(

2 2

n Rhc Z n

E

_n

= - C = -

! µ

3 4

4 c ! R e

p

=

K²

µ

Hydrogen isotopes

µ º m_eM / (m_e+ M)

isotope (mass) effect

H

_b

D

_b

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15/ 22

"Exotic" atoms

§ positronium (positronium) = (e

⁺

e

^-

) e

^-

e

⁺

§ muonium (muonium) (

^µ⁺

e

^-

)

v µ⁺

e

^-

§ meson atoms:

Similar potential Þ same level structure, Different µ Þ different energies

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muon atom (p µ ^- ):

orbit radius <R

_nucleus

Þ

the muon penetrates (probes) the nucleus

µ^-

p

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Hydrogen Atom in “4-D”

Laplace-Runge-Lenz vector (LRL-vector) Classical form : ⃗𝐴 = ⃗𝑝×𝐿 − 𝑚𝑘 ̂𝑟

LRL-vec. in an inverse-square central force.

It is evident that

^𝒅𝑨_𝐝𝐭

= 𝟎 Scaling this as 𝐷 =

_#!|5|³

.

Invariants for negative energies. 𝐷 ⋅ 𝐿 = 0 & 𝐷

^#

+ 𝐿

^#

=

^!$_#|5|^!

The Poisson brackets:

{𝐷

₆

, 𝐿

₇

} = 𝜀

_67$

𝐷

_$

{𝐿

₆

, 𝐿

₇

} = 𝜀

_67$

𝐿

_$

{𝐷

₆

, 𝐷

₇

} = 𝜀

_67$

𝐿

_$

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Quantum Mechanical Form: ^{⃗𝑝×𝐿} is non-Hermitian while

'

#

⃗𝑝×𝐿 − 𝐿× ⃗𝑝 is. Thus, the scaled LRL-vec becomes.

𝐷 = 1

2𝑚|𝐸

₊

| [ 1

2 ⃗𝑝×𝐿 − 𝐿× ⃗𝑝 − mk ̂𝑟]

With the commutation relations:

[𝐻

_8,

𝐷] = [𝐻

_8,

𝐿] = 0

[𝐿

_6,

𝐿

₇

] = 𝑖𝜀

_67$

𝐿

_$

Has SO(3) group structure.

[𝐿

_6,

𝐷

₇

] = 𝑖𝜀

_67$

𝐷

_$

[D

_:,

D

_;

] = iε

_:;<

L

_<

Together they give 𝑆𝑂(4) structure

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Explicit presence of SO(4)

Define: ℒ

^:;

≡ 𝜖

_67$

𝐿

_$

ℒ

_:(

≡ 𝐷

₆

Leads to [ℒ

_{:; ,}

ℒ

_:<

] = 𝑖ℒ

_;<

Casimir Invariants : 𝐿. 𝐷 = 0 ℒ ⃗

^#

= 𝐿

^#

+ 𝐷

^#

= −1 −

^'

#=_"

^^

Redefining the operators.

𝐽

_'

=

^'_#

(𝐿 − 𝐷 ), 𝐽

_#

=

^'_#

(𝐿 + 𝐷 ) we get the comm. relations 𝐽

_'6

, 𝐽

_'7

= 𝑖 𝜖

_67$

𝐽

_'$

𝐽

_#6

, 𝐽

_#7

= 𝑖 𝜖

_67$

𝐽

_#$

𝐽

_'6

, 𝐽

_#7

= 0

2 independent angular momenta are also constants

of motion.

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Generators

Generators for rotation : 𝑅 Ω = exp(−𝑖Ω ⋅ L) Where the 3 components of L transforms the 3 parameters of Ω and are thus the generators.

Transformation oper. for SO(4): 𝒯 ⃗𝛼, ⃗𝛽 = exp(−𝑖 ⃗𝛼 ⋅ L − ⃗𝛽 ⋅ ⃗𝐴)

= exp −𝑖 ⃗𝛼 𝐽

_'

+ 𝐽

_#

+ ⃗ 𝛽 𝐽

_#

− 𝐽

_'

= exp −𝑖 ⃗𝛼 − ⃗ 𝛽 𝐽

_'

exp −𝑖 ⃗𝛼 + ⃗ 𝛽 𝐽

_#

Where the isomorphism SO(4) ∼ SO(3) ⊗ SO(3) is evident.

And the Casimir Invariants in this case are

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From SO(3) (Quantum Angular Momentum).

We have 𝐽

_'^#

, 𝐽

_#^#

→ 𝐽 𝐽 + 1 , where 𝐽 = 0,

^'_#

, ⋯

This gives from the relation 2 𝐽

_'^#

+ 𝐽

_#^#

= −1 −

^'

=_"

𝐻

₈

= − 1

2 2𝐽

_'^#

+ 2𝐽

_#^#

+ 1 ⇒ 𝐸 = − 1

2(4𝐽 𝐽 + 1 + 1)

− 1

2𝑛

^#

= − 1

2 2𝐽 + 1

^#

Now as 𝐽

_'>

, 𝐽

_#>

can take values m

_:

∈ −𝐽, −𝐽 + 1, ⋯ , 𝐽 − 1, 𝐽

We have the degeneracy 2𝐽 + 1

^#

= 𝑛

^#

consistent with standard solution of the Schrödinger Eqn. if the el.

spin is not taken into account.

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Basis Vectors in 𝑱

_𝟏,𝟐^𝟐

_, 𝑱

_𝟏𝒁

_, 𝐉

_𝟐𝒁

𝐽

_'^#

𝐽, 𝑚1, 𝑚2 = 𝐽 𝐽 + 1 𝐽, 𝑚

_'

, 𝑚

_#

𝐽

_6>

𝐽, 𝑚1, 𝑚2 = 𝑚

₆

𝐽, 𝑚

_'

, 𝑚

_#

𝐿

_>

𝐽, 𝑚1, 𝑚2 = (𝑚

_'

+𝑚

_#

) 𝐽, 𝑚

_'

, 𝑚

_#

𝐷

_>

𝐽, 𝑚1, 𝑚2 = (𝑚

_'

−𝑚

_#

) 𝐽, 𝑚

_'

, 𝑚

_#

Note that the values

corresponding to 𝐿

_?

can take values between -2J and 2J = n-1.

2𝐽 = 𝑛 − 1 = 𝑛

_'

+ 𝑛

_#

+ |𝑀 | 𝑚

_'

+ 𝑚

_#

= 𝑀

𝑚

_'

− 𝑚

_#

= 𝑛

_'

− 𝑛

_#

Basis Vectors for spherical solution of the Schrödinger Eqn. 𝐽_! + 𝐽_" = 𝐿

Fundamentals of atomic physics