Lecture V Hydrogen Atom dr hab. Ewa Popko

1. Lecture V Hydrogen Atom dr hab. Ewa Popko

2. Niels Bohr1885 - 1962

3. Bohr Model of the Atom • Bohr made three assumptions (postulates) • 1. The electrons move only in certain circular orbits, called STATIONARY STATES. This motion can be described classically • 2. Radiation only occurs when an electron goes from one allowed state to another of lower energy. • The radiated frequency is given by • hf = Em - En • where Em and En are the energies of the two states • 3. The angular momentum of the electron is restricted to integer multiples of h/ (2p) = • mevr =n(1)

5. The hydrogen atom The Schrödinger equation Partial differential equation with three independent variables The potential energy in spherical coordinates (The potential energy function is spherically symmetric.)

6. The spherical coordinates (alternative to rectangular coordinates)

7. The hydrogen atom For all spherically symmetric potential-energy functions: ( the solutions are obtained by a method called separation of variables) Radial function Angular function of q and f Thus the partial differential equation with three independent variables three separate ordinary differential equations The functions q and f are the same for every spherically symmetric potential-energy function.

8. The solution The solution is determined by boundary conditions: - R(r) must approach zero at large r (bound state - electron localized near the nucleus); - Q(q) and F(f) must be periodic: (r,q,f) and(r,q,f+2p) describe the same point, so F(f)=F(f+2p); - Q(q) and F(f) must be finite. Quantum numbers: n - principal l – orbital ml - magnetic

9. Principal quantum number: n Ionized atom n = 2 - 3.4 eV n = 1 E = - 13.6 eV The energy En is determined by n = 1,2,3,4,5,…; n = 3 m - reduced mass

10. n= n=5 n=4 Energy Levels of the Hydrogen Atom Series limits E(eV) 0.00 -0.85 -1.51 n=3 Paschen -3.40 n=2 Balmer -13.6 n=1 Lyman

11. Balmer series 4000 5000 6000 7000 Å (Å) k(cm-1) HRed 6565 15234 Hgreenish-blue 4862 20565 Hblue 4342 23033 Hviolet 4103 24374 4000 5000 6000 7000 Å

12. Electron-Volt • There are many different units used to describe energy • One of the most useful in quantum physics is the electron-volt • The electron-volt is defined as the energy needed to move an electron through a potential difference of one volt 1(eV)=1.6*10-19J

13. The Rydberg constant • The Rydberg is a measure of energy normally expressed in m-1 but we can convert to other forms • R (J) = R(m-1)h c = R(eV) e wavelength

14. T. W. Hansch1941 - Rydberg measured (1998) to be R = 10 973 731.568 550 (84) m-1

15. Quantization of the orbital angular momentum. The possible values of the magnitude L of the orbital angular momentum L are determined by the requirement, that the Q(q) function must be finite at q=0 and q=p. Orbital quantum number There are n different possible values of L for the n th energy level!

16. Quantization of the component of the orbital angular momentum

17. Quantum numbers: n, l, m n – principal quantum number n – determines permitted values of the energy n = 1,2,3,4... l– orbital quantum number l -determines permitted values of the orbital angular momentum l = 0,1,2,…n-1; ml - magnetic quantum number ml – determines permitted values of the z-component of the orbital angular momentum

18. l = 0 n= 1 n= 2 l = 0,1 l = 1 m = ±1 n= 3 l = 0,1,2 yn,l,m Wave functions Q(q) - polynomial F(f) ~

19. Quantum number notation Degeneracy : one energy level En has different quantum numbers l and ml l = 0 : s states n=1 K shell l = 1: p states n=2 L shell l = 2 : d states n=3 M shell l = 3 : f states n=4 N shell l = 4: g states n=5 O shell . . . .

20. Electron states 3s K 1s 3p 2s M L 3d 2p