dan815
  • dan815
State space stuff @kainui http://prntscr.com/8qcmml
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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jamiebookeater
  • jamiebookeater
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anonymous
  • anonymous
horor question :v
dan815
  • dan815
ah its jut an intro question
Kainui
  • Kainui
Cool, looks like we just plug and chug into \[|\alpha|^2 + |\beta|^2 = 1\] for that first one.

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More answers

dan815
  • dan815
i think using alpha=x+iy = r*e^itan^-1(y/x)
dan815
  • dan815
|dw:1444631085836:dw|
Kainui
  • Kainui
Well here we're given that \[\alpha = e^{i \gamma} \cos \theta\] and beta as well
dan815
  • dan815
right okay so i just have to show it satisfies that eqn
dan815
  • dan815
it has to because that complex e^itheta is 1 rotated around so modulus of that =1
Kainui
  • Kainui
\[|\alpha|^2 = \alpha^* \alpha = e^{i \gamma - i \gamma} \cos^2 \theta = \cos^2 \theta\]
dan815
  • dan815
|dw:1444631358098:dw|
dan815
  • dan815
oo ok
dan815
  • dan815
i didnt think of just using conjugate like that
dan815
  • dan815
is this a property though? |A B| = |A| |B| ?
Kainui
  • Kainui
Well we could find out I guess
dan815
  • dan815
|AB|^2 = (AB)* BA = A* B* B A = |A| |B|
Kainui
  • Kainui
\[\sqrt{a^*ab^*b} = \sqrt{a^*a}\sqrt{b^*b}\] checks out
dan815
  • dan815
k cool
dan815
  • dan815
next qn
dan815
  • dan815
http://prntscr.com/8qcptc
dan815
  • dan815
what does it mean by same "physical" state
dan815
  • 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
Kainui
  • Kainui
it's the same physical state because a complex phase doesn't change it, \[|e^{i \theta} a|^2 = |a|^2\]
Kainui
  • Kainui
So yeah I think that's what you're saying
dan815
  • dan815
hmmm
Kainui
  • 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.
dan815
  • dan815
so like grapinh PSI has no meaning?
dan815
  • dan815
graphing*
Kainui
  • Kainui
Graphing anything has no meaning, however complex phase on a wave function has meaning and it's important to QM
Kainui
  • Kainui
It's just when you're looking at pure states that you don't have interference so it doesn't do anything haha
dan815
  • dan815
ok i see i guess multiplying by modulus 1 is just like multiplying some equation by 2 on both sides
dan815
  • dan815
y=x 2y=2x same thing
Kainui
  • Kainui
Not quite, the phase is more of a relative thing that gives you the interference patterns from being in two different states simultaneously
dan815
  • dan815
what does that mean
dan815
  • dan815
so since both states are rotated same equally
Kainui
  • 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.
dan815
  • dan815
its just the same sphere with a diff coordinate system
dan815
  • dan815
ah i see since probability didnt change it didnt change anything with the complex phase?
Kainui
  • 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.
dan815
  • dan815
okay i see this complex phase wrt to prob isnt really makign sense right now, ill come back to that then
Kainui
  • Kainui
The 'idea' of where a particle could have traveled is what interferes with the particle's path.
Kainui
  • 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.
dan815
  • dan815
oh right i do remember this its kinda weird
dan815
  • dan815
we even calculate the wave collision pattern
dan815
  • dan815
since we assume the particles is all places with some prob
dan815
  • 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
dan815
  • dan815
|dw:1444632321197:dw|
dan815
  • dan815
or maybe a phase just means the same wave pattern just happened somewhere else on the wall now
dan815
  • dan815
|dw:1444632415530:dw|