## dan815 one year ago State space stuff @kainui http://prntscr.com/8qcmml

1. anonymous

horor question :v

2. dan815

ah its jut an intro question

3. Kainui

Cool, looks like we just plug and chug into $|\alpha|^2 + |\beta|^2 = 1$ for that first one.

4. dan815

i think using alpha=x+iy = r*e^itan^-1(y/x)

5. dan815

|dw:1444631085836:dw|

6. Kainui

Well here we're given that $\alpha = e^{i \gamma} \cos \theta$ and beta as well

7. dan815

right okay so i just have to show it satisfies that eqn

8. dan815

it has to because that complex e^itheta is 1 rotated around so modulus of that =1

9. Kainui

$|\alpha|^2 = \alpha^* \alpha = e^{i \gamma - i \gamma} \cos^2 \theta = \cos^2 \theta$

10. dan815

|dw:1444631358098:dw|

11. dan815

oo ok

12. dan815

i didnt think of just using conjugate like that

13. dan815

is this a property though? |A B| = |A| |B| ?

14. Kainui

Well we could find out I guess

15. dan815

|AB|^2 = (AB)* BA = A* B* B A = |A| |B|

16. Kainui

$\sqrt{a^*ab^*b} = \sqrt{a^*a}\sqrt{b^*b}$ checks out

17. dan815

k cool

18. dan815

next qn

19. dan815
20. dan815

what does it mean by same "physical" state

21. dan815

i can see that multiplying by anything of modulus 1 will not change that state vector property* of the |w> = A|0> + B |1> |A|^2 + |B|^2=1

22. Kainui

it's the same physical state because a complex phase doesn't change it, $|e^{i \theta} a|^2 = |a|^2$

23. Kainui

So yeah I think that's what you're saying

24. dan815

hmmm

25. Kainui

I know this to be true, in fact I know I can always choose my eigenstates to be real and I can prove it.

26. dan815

so like grapinh PSI has no meaning?

27. dan815

graphing*

28. Kainui

Graphing anything has no meaning, however complex phase on a wave function has meaning and it's important to QM

29. Kainui

It's just when you're looking at pure states that you don't have interference so it doesn't do anything haha

30. dan815

ok i see i guess multiplying by modulus 1 is just like multiplying some equation by 2 on both sides

31. dan815

y=x 2y=2x same thing

32. Kainui

Not quite, the phase is more of a relative thing that gives you the interference patterns from being in two different states simultaneously

33. dan815

what does that mean

34. dan815

so since both states are rotated same equally

35. Kainui

The thing is, the particle has a probability of being in multiple states at once and those PROBABILITIES interfere with each other with the complex phase.

36. dan815

its just the same sphere with a diff coordinate system

37. dan815

ah i see since probability didnt change it didnt change anything with the complex phase?

38. Kainui

No no I think this is something I can't explain you have to hear Alan Adams explain it, I'm just a mortal.

39. dan815

okay i see this complex phase wrt to prob isnt really makign sense right now, ill come back to that then

40. Kainui

The 'idea' of where a particle could have traveled is what interferes with the particle's path.

41. Kainui

and because of this, the complex phase plays a role during this interference, but right now we're not doing anything complicated so let's just ignore this until we come to it and let's move forward.

42. dan815

oh right i do remember this its kinda weird

43. dan815

we even calculate the wave collision pattern

44. dan815

since we assume the particles is all places with some prob

45. dan815

i am kind of getting an idea, so i should be thinking of the wave patterns that can arise with some changes in this angle

46. dan815

|dw:1444632321197:dw|

47. dan815

or maybe a phase just means the same wave pattern just happened somewhere else on the wall now

48. dan815

|dw:1444632415530:dw|