Download 2.4. Quantum Mechanical description of hydrogen atom

2.4.
Quantum Mechanical description of hydrogen atom
Atomic units
Quantity
Ang. mom.
Mass
Charge
Permittivity
Length
Energy
Atomic unit
h̄
me
e
4πε0
a0 (bohr)
Eh (hartree)
SI
[J s]
[kg]
h [C] i
Conversion
h̄ = 1, 05459 · 10−34 Js
me = 9, 1094 · 10−31 kg
e = 1, 6022 · 10−19 C
C2
4πε0 = 1, 11265 · 10−10 Jm
C2
Jm
From these one can derive:
[m]
[J]
2
0 h̄
1 bohr = 4πε
= 0, 529177 · 10−10 m
me e2
2
1 hartree = 4πεe 0 a0 = 4, 359814 · 10−18 J
1 Eh ≈ 27, 21 eV
Eh ≈ 627 kcal/mol
The model of the hydrogen atom:
• an electron is „situated” around the nuclei which is not moving;
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• the interaction potential is given by the Coulomb interaction: V = − er
In quantum mechanics we have to solve the Schrödinger equation:
ĤΨi = Ei Ψi
with
• Ĥ is the Hamiltonian including the interactions within the system (kinetic and potential
energy): Ĥ = T̂ + V
• Ei is the total energy of the system
• Ψi (x, y, z) is the wave function describing the system, also called state function, here also
can be called the orbital of the electron.
Notes:
1. the energy is quantized;
2. i index denotes that there are several such states. The one with the lowest energy is called
the ground state, the others are the excited states.
About the solution: during the calculations it turns out that the states should not be labeled
with a simple index i, but rather with a triplet of numbers, the so called quantum numbers:
i → (n, l, m)
It also comes out from the calculation that quantum numbers can not have arbitrary values:
this is where the name is from! For the hydrogen atom the possible values of the quantum
numbers are:
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• n – principal quantum number : 1, 2, 3, . . ..
• l – angular momentum quantum number : 0, 1, 2, . . .(n − 1)
• m – magnetic quantum number : −l, − l + 1, . . ., 0, 1, . . ., l (2l+1 different values)
The quantum numbers are related to physical quantities:
• n: determines the energy: En = − 2n1 2 (Eh )
EXACTLY LIKE IN BOHR THEORY!!!
• l: determines the size of the angular momentum: |l| =
p
l(l + 1)(h̄)
• m: determines the z component of the angular momentum:
−l, −l + 1, ..., 0, 1, ...l
lz = m(h̄)
m =
What is the angular momentum?
Classical definition of the angular momentum:
L = r × p = mr × v
where m is the mass, v is the speed, p is the momentum, r is the position of the particle (see
figure).
Why is m called the magnetic quantum number?
m determines the z component of the angular momentum. Since the electron is moving
around the nuclei, and has a charge, it creates magnetic moment. There is a proportional
relation between angular momentum and magnetic moment:
µ = −µB l
µz = −m · µB
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where µB is a constant (called the Bohr-magneton).
How many different values m can have?
m = − l + 1, ..., 0, ..., l, i.e. 2l + 1 values.
Since the interaction with the magnetic filed will be proportional to the magnetic moment, its
magnitude depends on m. → in magnetic field the energy levels split up to 2l + 1 different values.
This is the so called Zeeman-effect.
l=0 → 1 energy level
l=1 → 3 energy levels
l=2 → 5 energy levels
etc.
Notation of the orbitals:
principal
quant. number (n)
1
ang. mom.
quant. number (l)
0
subshell
l
1s
magnetic
quant. number (m)
0
number of orbitals
on the subshell
1
2
0
1
2s
2p
0
-1,0,1
1
3
3
0
1
2
3s
3p
3d
0
-1,0,1
-2,-1,0,1,2
1
3
5
4
0
1
2
3
4s
4p
4d
4f
0
-1,0,1
-2,-1,0,1,2
-3,-2,-1,0,1,2,3
1
3
5
7
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Representation of the orbitals: 1s and 2s orbitals
There is a node on the 2s orbital, where the value of the wave function gets zero.
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Representation of the orbitals: 2p orbitals
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Representation of the orbitals: d orbitals
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Representation of the orbitals: dotting – the frequency of the dots represent the value: more
points mean larger value of the wave function.
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Radial electron density:
probability of finding the electron at distance r from the nuclei (i.e. in a shell of the spere).
Radial density for orbitals 1s, 2s and 2p:
Radial density for orbitals 3s, 3p and 3d:
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The spin of the electron
We want to prove that in the ground state of hydrogen atom l=0: we put it into the magnetic
field. We assume one beam:
This is the so called Stern-Gerlach experiment.
The beam of ground state hydrogen atom splits into two beams. This contradicts the theory,
since we have expected 1, 3, 5,. . . beams!
Conclusion:
• Pauli (1925): a „fourth quantum number” is needed;
• Goudsmit and Uhlenbeck suggested the concept of spin, as the „internal angular momentum”
Classically: if the electron is not a pointwise particle, it can rotate around its axis, either to the
right or to the left.
In quantum mechanics: the electron as a particle has „intrinsic” angular momentum, which is its
own property, like its charge.
What do we know about it?
• it is like the angular momentum since there is magnetic moment associated with it;
• its projection can have two different values.
p
magnitude: s(s + 1)h̄
s quantum number
z component: ms h̄
ms quantum number, or spin quantum number
ms = −s, −s + 1, . . . , s − 1, s
⇒ s = 12 since in this case ms = − 12 , + 12
FOR ELECTRONS s = 21 always!!!!!
The fourth quantum number is: m s
Thus, the electron has spin.
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What is spin?
Where it does originate from?
Bad question, we would not ask: why the electron has a charge?
Properties of the electron:
charge: −1
spin: 1/2
Spin is the intrinsic momentum of the electron.
Spectroscopic application: Electron Spin Resonance (ESR):
Summarized:
The states of the hydrogen atom are quantized and are characterized by quantum numbers:
n = 1, 2, . . .
l = 0, 1, . . ., n − 1
m = − l, − l + 1, . . ., l
ms = − 12 , 21
The energy depends only on n: En = −
1 1
2 n2 (Eh ).
There is a many-fold degeneracy!
In magnetic field the energy splits up according to the magnetic quantum number m.
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