calculations to converge

Lorentz Transformation:

simultaneous

To be consistent with the action of these operators the spherical harmonic wavefunctions should begiven the correct sign. This convention gives the appearance that harmonics for positive odd values of mhave their wavefunction multiplied by a factor of −1. ThusY1,1 = −√ 38π sin(θ )eiφ (2.17)Y1,0 =√ 34π cos(θ ) (2.18)Y1,−1 =√ 38π sin(θ )e−iφ (2.19)(2.20)17

ˆl+ |m〉 = ̄h√l(l + 1) − m(m + 1) |m + 1〉 (2.15)ˆl− |m〉 = ̄h√l(l + 1) − m(m − 1) |m − 1〉

Summary of Energy Levels3 integer quantum numbers are needed to describe the scalar solutions for the non-relativistic Hydrogenatom wavefunction (n, l, m).n = 0, 1, 2, 3 · · · 0 ≤ l ≤ n − l ≤ m ≤ +l. (2.13)The energies of the allowed states areEn = − 12e24πε0a01(n + 1)2 = − μe232π2ε20 ̄h21N2 = − RyN2 . (2.14)16

∃ xFx ⊢ ∃ x(Gx➝Fx)

⊢ (((P➝Q)➝(P➝R))➝(P➝(Q➝R)))

((S∨R)➝Q), ~(P∨~S) ⊢ ~(P⟷Q)

(T➝~P), (~T➝~P) ⊢ ~P

(P➝~P) ⊢ ~P

P, (R➝~Q), (P➝Q) ⊢ ~R

~~(P➝Q), P ⊢ Q

~P, (Q➝P) ⊢ ~Q

P, (P➝Q) ⊢ Q

If Thaddeus won’t study if wisdom isn’t a virtue, then wisdom is a virtue only on theassump@on that virtue makes for happiness

If Simon is saintly, then if Quirinius doesn’t fail to be punc@lious, Ruth reaps what shesows

If Paul doesn’t ponder, wisdom is a virtue only on the condi@on that virtue makes forhappiness

Ruth won’t reap what she sows provided that Quirinius fails to be punc@lious

If unjust conduct is good for the soul, then wisdom is a virtue only if virtue makes forhappiness

If unjust conduct isn’t bad for the soul, then virtue fails to make for happines

If Thaddeus studies, then it’s not the case that he doesn’t fail to study

The claim that virtue doesn’t makes for happiness is false

Only provided that wisdom is a virtue does virtue make for happiness

If Ruth reaps what she sows only if virtue fails to make for happiness, then wisdom is avirtue if unjust conduct isn’t bad for the soul

Wisdom is a virtue if virtue fails to not make for happiness

If Paul ponders, then Quirinius fails to be punc@lious

Paul doesn’t fall short of pondering

Simon is no saint

This prac7ce problem

Prac7ce Problem Set 2

Logic 1

Having cancer is a good thing, for whatever is required by a good thing is itself a goodthing. Being cured of cancer is a good thing, and being cured of cancer requires havingcancer.

If Sarah did the wiring, it was done right. If Marie did the plumbing, then it was doneright. As neither the wiring nor the plumbing were done right, Sarah didn’t do the wiringand Marie didn’t do the plumbing

If the budget is to be balanced, we will have to raise taxes or cut spending. If we cutspending, then en7tlement programs will have to be cut and Democrats will have a fit. Ifwe raise taxes, Republicans will go ballis7c. So, if the budget is balanced, Democrats willhave a fit and Republicans will go ballis7c.

likelihood L(D) = P (D|H)

Hypotheses