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PHY206: Atomic Spectra

PHY206: Atomic Spectra. Lectures 2 - 4. Lectures 2-4: Outline. Introduction Orbital Angular Momentum (1) Magnetic Moments Stern-Gerlach experiment: the Spin Proton, Neutron Magnetic Moments Example: Magnetic Resonance Imaging Orbital Angular Momentum (2)

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PHY206: Atomic Spectra

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  1. PHY206: Atomic Spectra Lectures 2 - 4

  2. Lectures 2-4: Outline • Introduction • Orbital Angular Momentum (1) • Magnetic Moments • Stern-Gerlach experiment: the Spin • Proton, Neutron Magnetic Moments • Example: Magnetic Resonance Imaging • Orbital Angular Momentum (2) • Operators of Orbital Angular Momentum • Angular Shapes of Particle Wavefunctions • Spherical Harmonics Atomic Spectra

  3. Quantum Mechanics • Particles in quantum mechanics are expressed by wavefunctions • Wavefunctions are defined in spacetime (x,t) • They could extend to infinity (electrons) • They could occupy a region in space (quarks/gluons inside proton) • In QM we are talking about the propability to find a particle inside a volume at (x,t) • So the wavefunction modulus is a Probability Density (probablity per unit volume) • In QM, quantities (like Energy) become eigenvalues of operators acting on the wavefunctions Atomic Spectra

  4. Operators Atomic Spectra

  5. The Internal Space So far we have been talking about the spatial wavefunction of a particle: However, in QM there is an internal space of degrees of freedom associated to a particle. These “internal” coordinates are independent of the spatial part: The ‘total’ wavefunction is the product of the spatial part times the rest of the internal space wavefunctions. Angular Momentum is one of these internal spaces. Atomic Spectra

  6. Orbital Angular Momentum Assume r and p are on the XY plane. Then z=pz=0 and L=Lz. This is not possible in Quantum Mechanics because it would mean: Atomic Spectra

  7. Angular Momentum Z-axis Z-axis Z-axis Heisenberg Uncertainty Principle prohibits the angular momentum L from having a definite direction in space! A projection in one axis can be measured (Lz by convention) but then nothing is known about L 3D direction: L is a fuzzy vector! Atomic Spectra

  8. Possible values for L and Lz The magnitude of the orbital angular momentum can take the values: When the magnitude is fixed by one of the possible quantum numbers l, then the angular momentum at any given direction (by convention, z direction) takes 2l+1 possible values: Atomic Spectra

  9. Angular Momentum (summary) Angular momentum is a vector. We would like to be able to determine the 3D orientation and length of such a vector. However, in Quantum Mechanics this is impossible. We can know the magnitude and one orientation (by convention the z-axis) simultaneously. The other orientations are completely unknown. Equivalent statement: The square of the magnitude (L2) and the z-component (Lz) simultaneously satisfy the same eigenfunctions. Atomic Spectra

  10. Magnetic Moments: classical magnets Magnetic moment proportional to the angular momentum Atomic Spectra

  11. Magnetic Moments: magnetic energies Energy of magnetic moment Inside a magnetic field B So, if we assume that a particle (or atom) has a magnetic moment, then if we place a number of these particles inside a magnetic field (say B field along the z axis), they will have energies: Assume B along the z axis Energy can take any value in this range The reason is that the magnetic moment mz can be anywhere between -m,+m Atomic Spectra

  12. Magnets in B fields Magnetic dipole in B-field Z decreasing B, dB/dz < 0 Y X If we send a classical magnetic dipole through a B field that changes with Z, it will experience a force that depends on the mz. If mz=0, it feels no force! Atomic Spectra

  13. Magnetic Moments: magnetic forces Our particles will actually “feel” a force if the B field is changing along Z. So, if we were able to send a beam of particles (let’s say Ag atoms) through an inhomogeneous B field, the atoms (tiny magnets) would feel a force proportional to their magnetic moment Z component (the one parallel to the B-field). For example those atoms with magnetic moment vector along Y, would not feel any force and propagate undeflected. z y e x Atomic Spectra

  14. Stern-Gerlach Experiment (1922) In 1922 Stern & Gerlach sent a beam of Silver atoms through an inhomogeneous B-field and detected their positions on a screen behind the magnet. Amazing Result: the beam is split in two bunches!!! (1922: birth of the spin) Atomic Spectra

  15. Gerlach's postcard (8 February 1922), to Niels Bohr. translation: “Attached is the experimental proof of directional quantization. We congratulate you on the confirmation of your theory.” B-field on: Two peaks! B-field off: No splitting Atomic Spectra

  16. Magnetic Moments: what went wrong? Reminder: for QM particles like the Ag atoms L is quantized: Atomic Spectra

  17. Magnetic Moments: what went wrong? Reminder: for QM particles like the Ag atoms L is quantized: So, the magnetic moment along Z can only take 1 or 3 or … 2l+1 discrete values, never 2 !!! Atomic Spectra

  18. Spin: a new quantum number The SG experiment revealed a half-integer quantum number: the spin. So, when we say angular momentum we mean: SPIN ORBITAL Atomic Spectra

  19. Magnetic moment of an atom Atomic moment is proportional to a g-factor which depends on l,s,j Lande g-factor for an atomic state with quantum numbers l,s,j Note that if: Ang. Momentum l = 0 then g=2 Spin s=0 then g=1 Atomic Spectra

  20. Proton , Neutron: magnetic moments Both proton and neutron have spin ½ (they are fermions). However they are composite particles, made of (roughly) 50% gluons and 50% quarks Proton mass Atomic Spectra

  21. Bohr Magneton, Nuclear Magneton For electrons the magnetic moments are always proportional to mB thus making it a fundamental unit for electron associated mang. moments. For Nuclei the magnetic moments are always proportional to mN thus making it a fundamental unit for Nucleon associated mangetic moments. • The energy difference between the two electron spin states in a magnetic field is 660 times larger than for proton spin states! • So it requires less energy for a ‘spin down’ proton in our body to flip to a ‘spin up’ state. • Our bodies have many unpaired protons in H2O. They can be detected using • Magnetic Resonance Imaging Atomic Spectra

  22. Mangetic Resonance Imaging (MRI) B0 B=0 DE = 2mzB B • Magnetic resonance imaging (MRI) depends on the absorption of electromagnetic radiation by the nuclear spin of the hydrogen atoms in our bodies. The nucleus is aproton with spin ½,so in a magnetic field B there are only two possible spin directions with definite energy. The energy difference between these states isDE=2mzB, withmz= 1.41 x 10-26 J /Tesla. Atomic Spectra

  23. Nuclear Spins and Moments Atomic Spectra

  24. Example 1: Proton energy splitting Assume a person is MRI scanned in magnetic field B=1 Tesla. What is the energy difference between proton up and down states? Answer: Atomic Spectra

  25. Example 2: MRI frequencies What is the frequency f of photons that can be absorbed by this energy difference? Answer: We apply energy conservation: Radio Frequency Atomic Spectra

  26. Example 3: angular momentum of a photon So, the protons in the human body absorb a photon and flip their spin from down to up. Find the angular momentum L of the absorbed photon. Answer: The angular momentum is conserved: So the angular momentum of the photon is 1: (it should have 3 possible polarizations) Atomic Spectra

  27. Orbital Angular Momentum (2) Classical Mechanics: Quantum Mechanics: introduce operators for position and momentum Atomic Spectra

  28. Spherical Symmetry of Wave Functions A particle without orbital angular momentum (L=0) has a spherically symmetric wave function (we will show this). This means that the probability to find this particle in space depends only on the 3D radius r. However, as soon as we introduce some oam say L=1, the particle wave function acquires an angular dependence Atomic Spectra

  29. Particles with Spherically Symmetric wave functions have L=0: But and since two parallel vectors have 0 x-product: This is true for all components of L, Lx, Ly, Lz and also L2. So spherically symmetric wave functions describe particles with zero orbital angular momentum Atomic Spectra

  30. Angular Shape of Wave Functions Any particle wavefunction can be expressed as a linear superposition of basis wavefunctions which may have simpler angular shapes. Basis for Position probability densities Atomic Spectra

  31. Eigenfunctions/values of (l=1 , m=0) Conclusion: the (1,0) wave function is: Atomic Spectra

  32. Eigenfunctions/values of (l=1 , m=±1) Conclusion: the (1,-1) wave function is: Conclusion: the (1,+1) wave function is: Atomic Spectra

  33. Summary • Orbital Angular Momentum is quantized in integer units of: • The orbital angular momentum of a particle is a fuzzy vector. • We can measure simultaneously only L and Lz • The components (Lx, Ly, Lz) do not commute • A particle with specific orbital angular momentum has a wavefunction with a specific angular shape: • L=0 gives a spherically symmetric wavefunction • L>0 gives a wavefunction with angular dependence Atomic Spectra

  34. Spherical Harmonics Ylm If we make the transformation to spherical coordinates r,q,f : We then can prove that the spherical harmonics are only functions of q and f and NOT r : Atomic Spectra

  35. Proof(1) Express the cartesian partial derivatives to spherical: Then replace, and obtain a very simple relation for Lz Atomic Spectra

  36. Proof(2) Instead of doing the same for Lx and Ly we calculate L+ L-: The reason is that L2 can be easily found as: So, both Lz and L2 are functions of q and f only Atomic Spectra

  37. Properties of the Spherical Harmonics Using the operators extracted in the previous page: So that the solution has the form: Hence the f q parts decouple Atomic Spectra

  38. Spherical Harmonics: the f part is simple The solution of the equation: Normalization So, the f part is simply an exponential while the q part gets more complicated for increasing l : Atomic Spectra

  39. The Expansion Theorem Since the spherical harmonics Y(q,f) form a complete set of orthonormal functions, any function y(q,f) can be expanded in terms of Y as follows: Angular integral Normalization Atomic Spectra

  40. What does this mean? is the probability that a simultaneous measurement of L2 and Lz on a particle described by the wavefunction y(q,f) gives: Question: what is the probability that a measurement of L2 will give ? Answer: it is just the sum of probabilities for each possible ml state: Atomic Spectra

  41. The Expansion Theorem A general wave function can be expressed as a Fourier series with basis functions the eigenfunctions of L2 and Lz . The Fourier coefficients are only functions of the radius r. r-dependent coefficients (partial waves in scattering theory) Eigenfunctions of L2,Lz (spherical harmonics) Atomic Spectra

  42. Examples of Spherical Harmonics Atomic Spectra

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