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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; 2 • 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: 18 • 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 19 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 20 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. 21 Representation of the orbitals: 2p orbitals 22 Representation of the orbitals: d orbitals 23 24 Representation of the orbitals: dotting – the frequency of the dots represent the value: more points mean larger value of the wave function. 25 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: 26 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. 27 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. 28