Lecture physics a2 angular momentum, atomic states, the pauli principle huynh quang linh

Lecture 11: Angular Momentum, Atomic States, & the Pauli Principle z z Parametric Curve 1.2 |Y00| y( t ) 1.2 1 1.2 Parametric Curve |Y10| 1.2 x( t ) |Y11| x10( t ) x11( t ) 0 1.2 2 Parametric Curve 1.2 1.2 z 1.2 1 Parametric Curve |Y20| 1.2 y10( t ) 1 1.2 1.2 Parametric Curve |Y21| 1.2 y11( t ) 1.2 0 1.2 1.2 2 y20( t ) 2 |Y22| x( t ) x21( t ) x20( t ) Parametric Curve 1.2 1.2 y21( t ) 1 1.2 1.2 y( t ) 1.2 Content  Schrödinger’s Equation for the Hydrogen Atom    Angular Momentum   Radial wave functions Angular wave functions Quantization of Lz and L2 Spin and the Pauli exclusion principle  Stern-Gerlach experiment  Nuclear spin and MRI Summary of s-states of H-atom  To get the exact eigenstates and energies for the “s-states” of the Coulomb potential, one needs to solve the radial SEQ   2 2  e2     m r r r  r  R( r )  ER( r )    Summary of wave functions for the “s-states”:  (spherically symmetric) The zeros in the subscripts below are a reminder that these are states with zero angular momentum 1 1 R10 f( x) 0.5 h( x) 0 0 0 rx 4a 4 R1,0 ( r )  e  r / a0 R20 0.5 d4( x) 00 000 R30 rx 10 10a 10  r   r / 2a0 R2,0 ( r )  1  e  2a0  000 rx 10 15 15a 15  2r  r    e  r / 3a0 R3,0 (r )     2  a0  3a0    Wavefunction of the H-atom  The solution to the Schrödinger Equation (SEQ) in spherical coordinates is a product wave function of the form: nlm ( r,q ,f )  Rnl ( r )Ylm (q ,f ) with quantum numbers: n l and m principal orbital q r x y z f magnetic (angular momentum) The Ylm(q,f) are known as “Spherical harmonics” They are related to the angular momentum of the electron Last lecture we studied the properties of the radial part Today we will examine the angular part Quantized Angular Momentum  The  dependence of our wavefunction must have the form Re(y) Yl ,m (q , f )  eim Reminder: eimf = cos(mf) + i sin(mf)  r f Semi-classical ‘argument’ for angular momentum quantization: An integer number, m, of wavelengths must fit around the circle (radius r) above! 2p r = ml m = 0, 1,  2,  3, m = ‘orbital’ magnetic quantum number Quantization arises because only certain wavelengths fit around path But, l = h/pt, where pt is the tangential momentum of ‘orbiting’ electron: pt r = Lz = m m = 0, 1,  2,  3, The angular momentum along a given axis can have only quantized values: Lz = 0, ,  2,  3, The “l” Quantum Number   The quantum number m reflects the component of angular momentum about a given axis L z  m where m  0,  1,  2, The quantum number l in the angular wave function Ylm(q,f) tells the total angular momentum L L2 = Lx2 + Ly2 + Lz2 is quantized The possible values of L2 are: L2  l( l  ) where l  , 1, , Summary of quantum numbers for the H-atom: Principal quantum number: n = 1, 2, 3, … Orbital quantum number: l = 0, 1, 2, …, n-1 Orbital magnetic quantum number: m = -l, -(l-1), … 0, … (l-1), l ( = ml ) Classical Picture of L-Quantization e.g., l = Lz L  l( l  )  2(  )   L = 6  +2 Classically, the angular momentum vector precesses around the z axis + - Lrp -2 Why can’t the orbital angular momentum vector simply point in a specific direction, e.g., along z? If it did, then that would mean that r and p must both be in the x-y plane But that means that there would be no position uncertainty Dz in the z-direction, nor any momentum uncertainty Dpz, i.e., Dz = Dpz =0, in violation of the Uncertainty Principle Probability Density of Electrons Probability density = Probability per unit volume = ynlm2  Rnl2 Ylm2 The density of dots plotted below is proportional to ynlm2 “1s state” “2s state” “2p states” n= 2 l = 0 ml = 0 0, ±1 The Angular Wavefunction, Ylm(q,)   The angular wavefunction may be written: Ylm(q,) = P(q)eim where P(q) are polynomial functions of q To get some feeling for these angular distributions, we make polar plots of the q-dependent part of |Ylm(q,)| (i.e., P(q)): (Length of the dashed arrow is the magnitude of Ylm as a function of q.) l =0 z Parametric Curve 1.2 q y( t )| |Y00 Y0,  4p l =1 1.2 q |Yx10(10t )|0 z z Parametric Curve Y10 y Y1,  cos q x 1.2 1.2 1 1.2 x( t ) z 1 1.2 1.2 1.2 y10( t )Curve Parametric z 1.2 Y00 q x |Y11|0 q Re{Y11} x11( t ) y y Y1,1  sin q 1.2 1 x The Angular Wavefunction, Ylm(q,) z l=2 z Parametric Curve 2 q Y20 |Yx20( 20t )|0 x Y2,  3 cos q  1 1.2 y z Parametric Curve 2 y20( t ) 2 q |Yx21(21t ) |0 Re{Y21} Y2,1  sin q cos q y x 1.2 Parametric Curve 1.2 1 1.2 y21( t ) q 1.2 z t) |Yx(22 |0 - + Re{Y22} Y2,  sin q x 1.2 1 1.2 y( t ) 1.2 - + y ... z f magnetic (angular momentum) The Ylm(q,f) are known as “Spherical harmonics” They are related to the angular momentum of the electron Last lecture we studied the properties of the radial part... Number   The quantum number m reflects the component of angular momentum about a given axis L z  m where m  0,  1,  2, The quantum number l in the angular wave function Ylm(q,f) tells the total... Schrödinger’s Equation for the Hydrogen Atom    Angular Momentum   Radial wave functions Angular wave functions Quantization of Lz and L2 Spin and the Pauli exclusion principle  Stern-Gerlach