Old and new quantum theory

Transcription

1 Old and new quantum theory Faults of the Bohr model: - gives only position of the lines and not the intensity - does not explain the number of electrons on each orbit - gives innacurate results for atoms with many electrons Correspondence law: The quantum description becomes a classic one for large quantum numbers.

2 De Broglie hypothesis The momentum of moving object corresponds to a wave L n m u e n r n n h 2 nl=2r The circumference of allowed orbit contains an integral number of de Broglie wavelengths

3 Wave properties of matter Davisson-Germer experiment: wave properties of electrons Thomson experiment: diffraction of electrons on thin polycrystalline foil Stern experiment: diffraction of hydrogen and helium atoms on crystals of lithium fluoride and sodium fluoride.

4 Heisenberg uncertainty principle Momentu and position cannot be precisely determined at the same moment. The particle can occupy level with well defined energy for a long time. The lifetime of massive particles is limited x p h 4 E t h 4

5 Heisenberg uncertainty principle

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7 Description of a wave Wave vector (wavenumber) Frequency and angular frequency Wave equation (propagation along x axis, positive direction) Differential equation f t x v 2 t 2 v 2 2 x 2

8 The electron wavefunction The wavefunction determines probability of finding the particle in a certain space (for a plane wave between x and x+dx) if the measurement was performed at time t P ( x, t) dx * dx 2 dx Density of probability

9 Schrödinger equation The wavefunctions are obtained by solving Schrödinger equation Stationary (time independent ) equation: Gradient Electron energy The potential in whichh the electron is The wavefunctions so called eigenfunctions of S.eq. - have to be: -finite -unequivocal -continuous The electron exists The values of the functions and their first derivatives are the same at the borders of areas with different potentials the transition between these areas must be smooth

10 Schrödinger equation potential step E>V 0 I V 0 II 0 Reflection R v v 1 1 B A * * B A 2 k1 k2 k k Transmition T v v R+T= =1 * 2C C 4 * 1A A 1 k 1 k 2 k k 2 2

11 Schrödinger equation potential step V E<V 0 I V 0 II 0 Classical: Area I v 1 2E m The electron does not penetrate area II Quantum Area I Area II C=0, Exponential decrease Electron penetrates area II, but the probability decreases exponentially

12 Finite barrier 2 T exp 2m V 0 E l Electron can pass through, despite lower energy. The probability decreases with barrier width.

13 Model of atom: infinite potential well Inside the well: Infinite well has rigid walls. The electron wavefunction must be a standing wave the waves afre reflected on the walls.

14 Quantum atom model Schrödinger equation is solved in spherical coordinates The solution can be separated in respect to the coordinates. Each coordinate depends on integer number so called quantum number.

15 Quantum numbers - principal quantum number (n = 1,2,3...) corresponds to the number of orbit from Bohr s model - azimuthal quantum number (l = 0,1,...,n 1) corresponds to the value of orbital angular momentum L (number of electron sub-shell) - magnetic quantum number (m l = l,..., 1,0,1,...,l) shows the projection of orbital angular momentum onto given axis. - spin quantum number S corresponds to the electron spin and equals 1/2. For any elementary particle, this is a constant value. - magnetic spin q. n. (m s = m,m = 1 / 2, 1 / 2) shows the direction of the spin J = L + S Wektory orbitalny i spinowy sumują się.

16 Spin and fine structure of emission lines The movement of electron around the core generates a magnetic field. This field interacts with electron magnetic moment, causing increase or decrease of total electron energy (spin-orbit interaction) The emission lines split into two lines (fine structure) Hydrogen fine structure Sodium doublet

17 How can we see the quantum numbers? Electron energy depends mainly on the principal number n E( n) m e Z n 0 2 e 4 1 The azimuthal q.n. is related to the angular momentum of electrons L Einstein de Haas experiment l l 1

18 How can we see quantum numbers? The electron energy in magnetic field depends on magnetic q.n. m (Zeeman effect). A similar effect is observed in strong electric field Stark effect E p B m B l Stern-Gerlach experiment: the total magnetic moment of silver atom equals the spin magnetic moment of single electron µ s (rest of the moments cancel each other). Two values of spin are possible: +1/2 i 1/2

19 How can we see quantum numbers? Stern-Gerlach experiment

20 Electron shells An electron shell is formed by all atomic orbitals representing the same principal number n Electron shells K,L,M,N,O,P,Q 2n 2 electrons on a shell

21 Principles of occupation of energy levels The occupation of energy levels minimizes the potential energy. Pauli exclusion: In an atom, two electrons cannot be characterized by the same quantum numbers : n, l, m l, m s Sommerfeld effect: States with different values of azimuthal q.n l are splitted. States with lower l are occupied first. Electrons on elyptical orbits with low l can get closed to the atomic core, than those on circular orbits with lower n. The states with high n and low l may represent lower energies, than those with low n and high l

22 Space distribution of probability: orbitals Energy depends mainly on the principal number n The wave functions are dependent on alll quantum numbers Degeneration: two or more wavefunctions represent the same energy. For each n, there is n different values of l.. For each l there is 2l+1 values of m Example of electron configuration: 1s 2 2s 2 2p 4 2 electrons with n=1,l=0 2 electrons with n=2,l= =0 4 electrons with n= =2,l=1

23 Orbitals shapes

24 Orbitals - shapes

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26 The principles of occupation of states

27 The periodic table

28 The periodic table Energy required for electron ionization (separation from atom).