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PHY206: Atomic Spectra
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  1. PHY206: Atomic Spectra Lectures 5 - 8

  2. Lectures 5-8: Outline • The Schrodinger equation in 3D • Introduction: Wave Equations • Separation of Variables • Radial Schrodinger Equation • The Hydrogen Atom • Energy Levels, Eigenfunctions • Angular Shapes • Quantum Mechanics of the Hydrogen atom • Radiative Transitions • Relativistic Effects Atomic Spectra

  3. Wave Equations Wave Equation for non-dispersive waves Solution Did you notice something? If you re-write the equation in “operator form” you see that the wave Y is an eigenfuction of an operator, with eigenvalue = 0 ! Wave Equation for non-dispersive waves Atomic Spectra

  4. A hint for relativistic Quantum Mechanics But the Einstein equation of Special Relativity is: Klein-Gordon Equation: A relativistic QM particle with mass m Atomic Spectra

  5. The non-Relativistic limit: We just saw that the equation of a freely moving Relativistic QM particle with mass m (without spin!) is: The non-relativistic limit of this equation is reached when we require the NR energy condition: Schrodinger Equation for a freely moving non-Relativistic QM particle Atomic Spectra

  6. Particle in a central potential The energy of a classical particle moving in a potential V(r) is : In Quantum Mechanics E is an operator acting on the particle wavefunction: If we replace the momentum and position with the corresponding operators: Atomic Spectra

  7. The Schrödinger Equation The Schrodinger Wave equation: V(x): Potential Energy Complex wave function Continuous functions Probability to find the particle at (x,t) Normalization Let Time independent Schrödinger equation Atomic Spectra

  8. Seperation of Variables in 1D Step1: time-dependent 1D Schrodinger wave equation: Step2: assume a free particle V(x)=0 and factorization Step3: find the solutions for the 2 differential equations Atomic Spectra

  9. Spherical Coordinates z q r y f x Polar angle Azimuthal Laplacian: Atomic Spectra

  10. Schrödinger Equation in Spherical Coordinates 3D Cartesian: Spherical coordinate system: Potential energy: Atomic Spectra

  11. Particle in a central potential V(r) Separate angular and radial parts by multiplying by r2: Angular Part But we know that the operator in the parenthesis is ~L2 : Atomic Spectra

  12. Particle in a central potential V(r) The Schrodinger equation now has separate Radial and Angular pieces : So we can write: But we know that: Atomic Spectra

  13. Particle in a central potential V(r) So the Ylm now can be removed: If we absorb the r in the derivative of R and divide by r2 : Obviously the following simple redefinition leads to the Radial Schrodinger equation: Atomic Spectra

  14. Radial Schrodinger Equation: … leads to the Radial Schrodinger equation: This is basically a motion of a particle with angular momentum L in an effective 1D potential given in the parenthesis. Application: the Hydrogen atom For finite R at r=0 we have u(r=0)=0, also u(r)=0 at infinity Atomic Spectra

  15. H-atom: Energy Levels Effective Potetial The minimum of the effective potential is at: Bohr Radius Rydberg Energy Atomic Spectra

  16. Physical Consequences For l=0 the potential minimum tends to –infinity which physically means that our electron in the H-atom should fall in the nucleus. Actually this doesn’t happen in QM due to the Heisenberg Uncertainty principle: The more we localize a particle the higher its kinetic energy (fluctuations). The minimum kinetic energy is given by the uncertainty principle. Atomic Spectra

  17. Significance of the Bohr radius Now add this extra kinetic energy to the effective potential and find a new approximate minimum for particles with l=0: So the lowest possible state for an electron with l=0 in a Coulomb potential has an average radius equal to the Bohr radius: the electron will not fall in the nucleus! Atomic Spectra

  18. Energy Levels and Radial Eigenfunctions u(r)0 as rinfinity Polynomial with nr zeros: nr denotes the number of nodes between 0 and inf. u(r)0 as r0 Atomic Spectra

  19. H Atom: quantum numbers Principal quantum number n=1, 2, 3 … Energy where E0=13.6eV Orbital quantum number l=0, 1, 2, 3 …, n-1 Orbital angular momentum Magnetic quantum number m=-l, -l+1, …, l-1, l Angular momentum in Z-direction • Bound state energies are negative. • Energy dependence only on n is a result of spherical symmetry. • Angular momentum is quantized in magnitude and direction Atomic Spectra

  20. Include spin: J=L+S As we will see later the spin of the electron in the Hydrogen atom causes a small (about 1/1000) but measurable effect in the energy levels. The total angular momentum J is then given by the sum of the orbital part L and the spin part S. Spectroscopic Notation: Atomic Spectra

  21. H atom: the ground state wavefunctions Ground state n=1, l=m=0 where Bohr radius Normalization condition Radial probability density Electrons can be anywhere, but most likely to be at r=a0for the ground state. Atomic Spectra

  22. H atom: first excited state wavefunctions 1st excited (n=2) state has three degenerate states n=2, l=0, m=0 n=2, l=1, m=0 n=2, l=1, m=±1 Bohr model: Atomic Spectra

  23. Angular Shape of Wave Functions “2s state” “2p states” “1s state” n = 1 2 2 l = 0 0 1 ml= 0 0 0, ±1 Atomic Spectra

  24. Parity Imagine an operator P (called Parity) with the property: The eigenfuctions and eigenvalues of P can be found: So we can only have two eigenvalue equations: Parity + (even function) Parity - (odd function) Atomic Spectra

  25. Parity For a particle in a central potential: When l = 0,2,4,… (even) then the parity is even (+) When l = 1,3,5,… (odd) then the parity is odd (-) Conservation of Parity: Formal: when the parity operator P commutes with the Hamiltonian Meaning: no observable change in the state of a particle when the coordinates undergo a reflection through the origin Example: Electromagnetic Interactions conserve parity (i.e. they don’t care about reflections!) Atomic Spectra

  26. Radiative Transitions n=1 n=3 n=4 n=2 S P D F G l=0 1 2 3 4 n=5 Degenerate states Principal quantum number n=1, 2, 3 … with E0=13.6eV Determines energy Atomic Spectra

  27. Interaction with a field Interaction with a ‘field’ or potential V: No Interaction 1 Interaction 4 Interactions All we need is the probability to go from the initial state, to the final: Atomic Spectra

  28. Electric Dipole Transitions The hydrogen atom consisting of a proton and an electron is an electric dipole which creates an EM field or a potential: To find transition probabilities between n,l,m and n’,l’,m’ we use the Parity operator 0 for Not 0 for Atomic Spectra

  29. Spontaneous Transitions: the vacuum is not empty Fields and potentials are generated by (source) particles. For example the presence of a proton in space creates an EM field extending to infinity. What happens if we remove all of these particles and their fields? Vacuum: i.e. empty space However, QM allows vacuum energy to fluctuate according to the Heisenberg Uncertainty Principle: electron positron So, for a sufficiently small time interval Dt we can have enough energy to produce from … “nothing” an electron positron pair. The e-p pair will annihilate to “nothing” after Dt! Atomic Spectra

  30. Spontaneous Transitions So, the vacuum is boiling by having particles continuously produced and destroyed. These particles produce a real field that can interact with the Hydrogen atom. This interaction can cause transitions from n to n’ with n>n’: Lyman series Balmer series Atomic Spectra

  31. The Reduced Mass Effect So far we have been assuming MNucleus >> me . The effects of a finite MN are taken into account by considering the reduced mass: Atomic Spectra

  32. Relativistic Effects We now assume p<<mec and expand the square root about 1: Leading relativistic correction Non-Relativistic Kinetic Energy Rest Mass Atomic Spectra

  33. The fine structure constant The average momentum of an electron in the hydrogen atom can be estimated by the uncertainty principle Bohr Radius Fine Structure Constant = So, we can use this estimate for the leading relativistic correction: Relativistic correction to the kinetic energy of the electron in the H-atom Atomic Spectra

  34. The spin-orbit interaction As we have seen electrons have spin S and a magnetic moment given by: But the electron also spins around the proton creating a magnetic field: That gives rise to an interaction of the electron spin to the B-field and an energy: Atomic Spectra

  35. Effect of the SL interaction on the hydrogen atom energy states The energy due to the spin-orbit interaction for a low lying H-atom state is of order: Where constant f ~ 1/10 Fine Structure Constant = with So, we expect the spin-orbit interaction to be an effect ~1000 times smaller than the ER/n2 levels we extracted from the Schrodinger equation. Atomic Spectra

  36. Examples: n=2, l=0,1 and s=1/2 Spectroscopic Notation: We can experimentally verify the spin orbit effect by measuring an energy difference of: by looking at small difference in the wavelengths of the transitions: Atomic Spectra