(n, l, m l ) 3/2/2016. Quantum Numbers (QN) Plots of Energy Level. Roadmap for Exploring Hydrogen Atom

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1 PHYS 34 Modern Physics Atom III: Angular Momentum and Spin Roadmap for Exploring Hydrogen Atom Today Contents: a) Orbital Angular Momentum and Magnetic Dipole Moment b) Electric Dipole Moment c) Stern Gerlach Experiments d) Spin Angular Momentum and Addition of Angular Momentum Today Lecture Quantum Numbers (QN) Quantum Numbers Values that emerge from the wave functions of Schrodinger Equation 1 st n = energy level QN (n=1,,3..) nd l= orbit angular momentum QN (l=0,1, n 1) 3 rd m l = magnetic QN (m l = l, l+1,..0 l 1,l) Plots of Energy Level (n, l, m l ) For the same n but different l and m l, the energy levels have different orbital angular momentum, but they all have the same energy. It is called energy degeneracy 1

2 Plots of Energy Level (n, l, m l ) Can we make the energy levels of the same n but different l and m l to have different energy? It is called energy splitting. Adding a Magnetic Field A PHY 51 Question: What is a magnetic dipole moment for a current i flows around a loop of radius r? Yes, we can! Add magnetic fields or electric fields. A PHY 51 Question: What is the current of i if an electron e moves around a loop of radius r with a linear momentum of p=mv? ICP 6: Prove that an electron induces a magnetic dipole momentum by Magnetic Dipole Momentum and Angular Momentum An electron induces a magnetic dipole momentum by The magnetic dipole momentum of an moving electron relates to its angular momentum by, Magnetic Dipole Momentum and Angular Momentum, Define. /, we have,

3 Electric Dipole Moment Hydrogen atom has positive charge (proton) and negative charge (electron) separated in space, so it also has electric dipole momentum,. is the distance between the electron and the nuclei.,, Electric Dipole Moment 0,? 0, 0 The non uniform electric field has a non zero net force on the electric dipole momentum. Force on Magnetic Dipole Moment Stern-Gerlach Experiments, 0, 0 the n=1 state of hydrogen atom The non uniform magnetic field has a non zero net force on the magnetic dipole moment. For an atom,, What happened if I send in the group state hydrogen atom?, 3

4 Stern-Gerlach Experiments Stern-Gerlach Experiments??? What happened if I send in the group state hydrogen atom?, What happened if I send in the group state hydrogen atom?,????!!! Stern-Gerlach Experiments??? Spin Angular Momentum For the group state hydrogen atom,,wehave splitting due to spin: What happened if I send in the group state hydrogen atom?,?!!! We add some intrinsic angular momentum to explain the observation, and called these intrinsic angular momentum to spin angular momentum or spin. 4

5 What is Spin? Spin is one of the quantum mechanical properties of fundamental particles. It borrows the name of spin from classical physics, but it is not a spinning motion of electron. It is an intrinsic property just like charge, mass, isospin,.. Quantum Numbers (QN) Quantum Numbers Values that emerge from the wave functions of Schrodinger Equation 1 st n = energy level QN (n=1,,3..) nd l= orbit angular momentum QN (l=0,1, n 1) 3 rd m l = orbit magnetic QN (m l = l, l+1,..0 l 1,l) 4 th S = spin angular momentum QN (S = for one electron) 5 th m s = spin magnetic QN (m s = s, s+1,..0 s 1,s) Addition of Angular Momentum Quantum Numbers Addition of Angular Momentum L tot = L 1 + L L tot = L 1 + L L L is vector: In classical physics, L tot can be any value between L 1 + L and L 1 - L. L 1 tot m tot, Y ltot m tot 1 m 1, Y l1 m 1 m Y l m, However, in quantum mechanics, L tot is not only a vector but also represents the quantization number of angular wave function. L tot must be integers. 5

6 Addition of Angular Momentum L tot = L 1 + L ICP 7: The addition of two angular momentum. Fill the question marker in the following table. l 1 =1, l =1, m s degenerate tot m tot 1 m 1 m m 1 m m tot m tot Possibilities (m 1,m )?? 1 tot 1 m tot m 1 m??? Addition of Spin and Orbital The addition of spin and orbital is called the total angular momentum, J. Vector Representation of J J = L + S J s j s S m m j m s L L=, S=1/ J =5/ or 3/ jand m j are our 6 th and 7 th quantum number. 6

7 Picture of L-S Addition How to Label an Atomic State Now we have seven quantum numbers for a certain atomic state. ( n, l, m l, s, m s, J, m j ) energy orbital level z orbital spin z spin total z total Pictures for thinking the addition,,, Sometimes we can ignore s+1 for hydrogen More Detailed H-atom Level Scheme ICP8 3s 3p 3d 3p 3/ 3s1/ 3p 1/ 3d 5/ 3d 3/ 3s 3p 3d? 3p 3/ 3s1/ 3p 1/ 3d 5/ 3d 3/ p 3/ p 3/ s p s1/ p 1/ s p s1/ p 1/ 1s 1s 1/ 1s 1s 1/ Energies & Spectra not sensitive to lm l Energies & Spectra sensitive to j m j lm l s m s Energies & Spectra not sensitive to lm l Energies & Spectra sensitive to j m j lm l s m s 7

8 HW8 HW7: Chapter 7- Problem 1,14,19,0,5,7 and ICP6-30 8