Physics 3313 - Lecture 15 Wednesday March 24, 2010 Dr. Andrew Brandt Hydrogen Atom HW 6 on Ch. 7 is assigned due Weds 3/31 3/24/2010 3313 Andrew Brandt
CHAPTER 7 The Hydrogen Atom 7.3 Quantum Numbers 7.4 Magnetic Effects on Atomic Spectra – The Normal Zeeman Effect 7.5 Intrinsic Spin 7.6 Energy Levels and Electron Probabilities 3/24/2010 3313 Andrew Brandt
7.3: Quantum Numbers The three quantum numbers: n Principal quantum number ℓ Orbital angular momentum quantum number mℓ Magnetic quantum number The boundary conditions: n = 1, 2, 3, 4, . . . Integer ℓ = 0, 1, 2, 3, . . . , n − 1 Integer mℓ = −ℓ, −ℓ + 1, . . . , 0, 1, . . . , ℓ − 1, ℓ Integer The restrictions for quantum numbers: n > 0 ℓ < n |mℓ| ≤ ℓ 3/24/2010 3313 Andrew Brandt
Hydrogen Atom Wave Function http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hydwf.html#c1 3/24/2010 3313 Andrew Brandt
Principal Quantum Number n It results from the solution of R(r). The result for this quantized energy is 3/24/2010 3313 Andrew Brandt
Orbital Angular Momentum Quantum Number ℓ It is associated with the R(r) and f(θ) parts of the wave function. Classically, the orbital angular momentum with L = mvorbitalr. ℓ is related to L by . In an ℓ = 0 state, . This disagrees with Bohr’s semi-classical “planetary” model of electrons orbiting a nucleus L = nħ. 3/24/2010 3313 Andrew Brandt
Orbital Angular Momentum Quantum Number ℓ A certain energy level is degenerate with respect to ℓ when the energy is independent of ℓ. Use letter names for the various ℓ values. ℓ = 0 1 2 3 4 5 . . . Letter = s p d f g h . . . Atomic states are referred to by their n and ℓ. A state with n = 2 and ℓ = 1 is called a 2p state. The boundary conditions require n > ℓ. 3/24/2010 3313 Andrew Brandt
Magnetic Quantum Number mℓ The angle is a measure of the rotation about the z axis. The solution for specifies that mℓ is an integer and related to the z component of L by Only certain orientations of L are possible and this is called space quantization 3/24/2010 3313 Andrew Brandt
Orbital (l) and Magnetic (ml) Quantum Numbers l is related to orbital angular momentum; angular momentum is quantized and conserved, but since h is so small, often don’t notice quantization Electron orbiting nucleus is a small current loop and has a magnetic field, so an electron with angular momentum interacts with an external magnetic field The magnetic quantum number ml specifies the direction of L (which is a vector—right hand rule) and gives the component of L in the direction of the magnetic field Lz Five ml values for l=2 correspond to five different orientations of angular momentum vector. 3/24/2010 3313 Andrew Brandt
Magnetic Quantum Number mℓ Quantum mechanics allows L to be quantized along only one direction in space. Because of the relation L2 = Lx2 + Ly2 + Lz2 the knowledge of a second component would imply a knowledge of the third component because we know L . We expect the average of the angular momentum components squared to be 3/24/2010 3313 Andrew Brandt
Angular Momentum L cannot be aligned parallel with an external magnetic field (B) because Lz is always smaller than L (except when l=0) In the absence of an external field the choice of the z axis is arbitrary (measure projection as in any direction) Why only Lz quantized? What about Lx and Ly? Suppose L were in z direction, then electron would be confined to x-y plane; this implies z position is known and pz is infinitely uncertain, which is not true if part of a hydrogen atom 3/24/2010 3313 Andrew Brandt
Precession of Angular Momentum The direction of L is thus continually changing as it precesses around the z axis (note average values of Lx =Ly =0 ) Therefore and it is only necessary to specify L and Lz 3/24/2010 3313 Andrew Brandt
Angular Momentum Operator Consider angular momentum definition: so We can define the angular momentum operator in cartesian and spherical coordinates: with gives similarly 3/24/2010 3313 Andrew Brandt
(Normal) Zeeman Effect The Zeeman effect is the splitting of spectral lines into separate sub-lines when atoms radiate in a magnetic field , with a spacing of the lines dependent on the strength of the magnetic field (first observed by Peter Zeeman in 1896). Can this be explained by Quantum Mechanics? Start the journey by considering a particle with mass m and charge e moving with a speed v in a circular orbit of radius r , we define its magnetic moment as We would like to relate the magnetic moment to the angular momentum L=mvr d=vt so if the particle traverses a circle once in a time t=T=1/f (where f, the frequency, is the inverse of the period T); it travels a distance 2r so and so why negative? for a positron but for an electron they are anti-parallel 3/24/2010 3313 Andrew Brandt
Magnetic Moment What about the magnitude of the magnetic moment? The Bohr magneton is defined as And the z component? 3/24/2010 3313 Andrew Brandt
Torque and Energy What happens if you put a magnetic moment in a magnetic field? Yup: maximum torque when effect is to rotate dipole, therefore a dipole has energy due to its position in a magnetic field. By convention B is in the +z direction: Combine with gives Energy of atomic state depends on magnetic quantum number (not just n) if atom is in a magnetic field ! 3/24/2010 3313 Andrew Brandt
Zeeman Effect So since ml has 2l+1 different values, a given state with angular momentum l will split into 2l+1 different sub-states, each separated in energy by The explanation of the Zeeman effect is one of the triumphs of quantum theory: 3/24/2010 3313 Andrew Brandt
Zeeman Effect Note for l=2 get splitting into 5 lines, but due to selection rules, still only 3 different energy photons 3/24/2010 3313 Andrew Brandt
Anomalous Zeeman Effect The Normal Zeeman effect is the splitting of spectral lines into 2l+1 separate sub-lines giving 3-fold splitting of spectral lines Some elements did not behave this way but split into different numbers of sub-lines “Pauli why do you look so sad?” “How can one look happy when he is thinking about the Anomalous Zeeman effect!” Recall and So electrons with different ml have different energy What if you sent a beam of atoms in different ml states through a magnetic field They would experience a force proportional to ml (if magnetic field not uniform) 3/24/2010 3313 Andrew Brandt
The Normal Zeeman Effect An atomic beam of particles in the ℓ = 1 state pass through a magnetic field along the z direction. The mℓ = +1 state will be deflected down, the mℓ = −1 state up, and the mℓ = 0 state will be undeflected. If the space quantization were due to the magnetic quantum number mℓ, mℓ states is always odd (2ℓ + 1) and should produce an odd number of lines. 3/24/2010 3313 Andrew Brandt
Stern-Gerlach Experiment Used silver atoms in ground state: l=0 so ml=0 Expect 1 line Observe 2 lines 3/24/2010 3313 Andrew Brandt
Spin Intrinsic angular momentum of electron independent of orbital angular momentum Like revolution and rotation 2s+1=2 -> s=1/2 (always ½ for electron) 3/24/2010 3313 Andrew Brandt
4th Quantum Number So spin is the 4th quanutum number Two 1s ground states and In general probability of measuring ms=+1/2 in Stern-Gerlach is |C2| for unpolarized beam |C2|= |C’2|=1/2 3/24/2010 3313 Andrew Brandt
Spin Magnetic Moment Recall g=gyromagnetic ratio = -e/2m For spin and gs/gl~2 deviations from 2 could indicated new physics g-2 experiments 3/24/2010 3313 Andrew Brandt