State space stuff @kainui http://prntscr.com/8qcmml

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State space stuff @kainui http://prntscr.com/8qcmml

Mathematics
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horor question :v
ah its jut an intro question
Cool, looks like we just plug and chug into \[|\alpha|^2 + |\beta|^2 = 1\] for that first one.

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