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Transcript
General properties of radiative transitions: - Bohr's postulate for the transition frequency: - consider a general solution ψn to a time-independent Schrödinger equation corresponding to the energy eigenvalue En - position expectation value: - <x> is time independent and thus the average position of the electron does not change in a given state. Therefore, the electron does not radiate as long as it remains in a specific eigenstate ψn of the Hamiltonian describing it. - Consider an electron changing its quantum mechanical state from ψn to ψm, i.e. changing its energy from En to Em. During the transition the electron is in a state phys4.8 Page 1 - probabilities: - initial state n: - final state m: - radiation is generated during the transition from n to m - calculate the position expectation value <x> in an intermediate superposition state phys4.8 Page 2 - the expectation value <x>mn oscillates at the transition frequency - <x>mn is also called the transition matrix element - the strength of the transitions, i.e. the brightness of a spectral line in emission or the strength of absorption, is proportional to the transition matrix element - transitions for which <x>mn ≠ 0 are called allowed transitions, transitions for which <x>mn = 0 are called forbidden transitions - the conditions on two states with quantum numbers n and m that lead to allowed transitions are called selection rules phys4.8 Page 3 Selection rules for transitions in the Hydrogen atom: - w.f. with characteristic quantum numbers - allowed transitions - for u = x the dipole emitting the radiation would be oriented along the x-direction - solve integral for all transitions - allowed transitions phys4.8 Page 4 interpretation of selection rules: - angular momentum selection rule - the photon emitted carries an angular momentum ± h/2π so that the total angular momentum is conserved in a transition - magnetic quantum number selection rule - right circularly polarized light (σ+) - linearly polarized light (π) - left circularly polarized light (σ-) - classically the analog of a photon with angular momentum ± h/2π would correspond to a left or right circularly polarized electromagnetic wave phys4.8 Page 5 Zeeman Effect - consider the potential energy of a magnetic moment μ in an external field B - the minimum energy configuration U = - μ B occurs when μ is aligned parallel to B, the maximum energy U = + μ B occurs when μ is anti parallel to B. - a current I flowing in a loop enclosing the area A generated by a single electron with a orbit period T = 1/f generates a magnetic moment along the direction n normal to the area - this is related to the electrons angular momentum - thus phys4.8 Page 6 - with the gyromagnetic ratio of the electron given by: - thus the potential energy of an electron with an angular momentum of L in an external field B is given by - quantum mechanically the z component lz of L is given by - there fore in the hydrogen atom - with the Bohr magneton phys4.8 Page 7 Spectrum of Hydrogen in a Magnetic Field - the splitting of degenerate spectral lines (at the same energy) into individual lines by magnetic field is called Zeeman effect after Pieter Zeeman who discovered it in 1896 - 2 l + 1 different values for ml result in the same number of levels with different energy - number of transition frequencies is limited by selection rules Δl =± 1, Δml = 0, ± 1 - in the normal Zeeman effect three different lines occur phys4.8 Page 8 Electron Spin: The solution of the spatial part of the hydrogen Schrödinger equation is not sufficient to describe all properties of the hydrogen spectrum. An additional property of the electron, i.e. the electron spin, needs to be considered. - example: splitting of spectral lines into closely spaced pairs (doublets) - e.g. first line in Balmer series (n=3 -> n=2) at λ = 656.3 nm is split into two lines with separation Δλ = 0.14 nm in absence of external field - such a splitting in the transition frequency is called fine structure normal Zeeman effect: transition frequency split into three lines anomalous Zeeman effect: splitting into 4 or 6 or even more lines with separation different from Zeeman prediction phys4.8 Page 9 Goudsmit and Uhlenbeck: Every electron has an intrinsic angular momentum S, called spin. The electron spin is associated with a magnetic moment μ. - their idea was motivated by considering a spinning electron with charge e giving rise to a magnetic moment and a spin angular momentum - considering the size of the electron (re < 10-16 m) the rotation rate would need to be so high that the equatorial velocity would be much greater than the velocity of light - no classical explanation for the phenomenon of spin Relativistic Quantum Mechanics (Paul Dirac): A particle with mass and charge of an electron must have the observed intrinsic angular momentum and magnetic moment. Spin quantum number: the electron spin quantum number is s = 1/2. Spin angular momentum: - note that the relation between the spin angular momentum and its quantum number is identical to the one for the orbital angular momentum L and its quantum number l phys4.8 Page 10 Spin and space quantization: spin: z-component of spin: - spin magnetic quantum number ms - i.e. 2 s + 1 = 2 different orientations of spin with respect to quantization axis (e.g. an magnetic field Bz along z-direction) - spin magnetic moment of the electron - magnetic moment of the electron along z-direction - Bohr magneton phys4.8 Page 11 - gyromagnetic ratio for orbital motion of electron: - gyromagnetic ratio for electron spin: - the electron g-factor - g-factor is predicted to be g = 2 from relativistic quantum mechanics - the small correction is due to quantum electrodynamics (QED) effects (vacuum fluctuations) - full set of electron quantum numbers: energy orbital angular momentum orbital angular momentum direction spin direction phys4.8 Page 12 Stern-Gerlach experiment: - neutral silver (Ag) atoms: magnetic moment determined by single electron - in an inhomogeneous magnetic field a force acts on a magnetic moment depending on its orientation - a homogeneous field generates only a torque - potential energy of atom magnetic moment - force on magnetic moment of atom phys4.8 Page 13 - classically all magnetic moment (spin) orientations should occur, but only two orientations corresponding to the two possible spin states of the electron are observed - first direct evidence for directional quantization (1921), there are only discrete possibilities for the orientation of the magnetic moment with respect to the external field. - quantitative analysis allows to measure value of the Bohr magneton - general method to directly measure magnetic moments of atoms - All atoms with one s electron show the same displacement. This indicates that the magnetic moment of all electrons in closed inner shells is canceled out. - For an outermost s electron there is no orbital angular momentum. Therefore the spin angular momentum is directly measured. - the magnitude and direction of angular and spin moments is conserved when atoms move in space phys4.8 Page 14