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Taking first semester Quantum Mechanics and our first assignment is some pretty basic stuff related to the braket notation.
I'm realizing I was never very fluent in my complex conjugates and how to deal with absolute values (yikes!).
I'm wondering if there are some good examples anywhere online regarding normalization of a wave state? Or even somewhere I can check my work to make sure I'm correct.
For example, here is one of my problems:
[math]\mid \psi \rangle = 3\mid + \rangle - e^{i \pi /3} | - \rangle [/math]
[math]1=\langle \psi \mid \psi \rangle[/math]
[math]1= C^* ( 3\langle+| + e^{i\pi/3} \langle-|)C(3\langle+| - e^{i\pi/3} \langle-|)[/math]
And in previous problems I end up with |C|^2= -1/7, which doesn't make sense.
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>>9160391
Bras and kets are complex conjugates of each other, so just take the complex conjugate of the ket then slam them together and solve.
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>>9160399
I know. But what am I doing wrong here? How can the absolute value of C^2 be negative??
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>>9160410
Your bra has picked up a minus on the 4, but 4 isn't complex. [eqn] | \psi \rangle = 3 | + \rangle + 4 | - \rangle \\ \implies \langle \psi | = 3 \langle + | + 4 \langle - | [/eqn]
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>>9160417
But I thought the bra was the conjugate? Hmm, something I'm not getting here. I mean it makes sense, so you only switch the signs if the 'b' term in the complex number is imaginary?
What about this, pic realted.
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>>9160427
When you take the complex conjugate of something the sign attached to the [math] i [/math] changes, so [eqn] \left ( e^{ i \pi / 3 } \right ) ^{*} = e ^ { -i \pi / 3 } [/eqn]
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http://www.mathpages.com/home/kmath638/kmath638.htm
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|+> and |-> are orthonormal. <+|->=0, <+|+>=1, etc.
<psi|psi>=9+(-exp(-ipi/3))(-exp(ipi/3))=9+1=10
So the normalized state is |psi>/sqrt(10)
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>>9160438
>>9160427
Okay so... this!
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Don't know if this is useful at all, but here goes
[math] \displaystyle
\\ z_1 = x_1+y_1i \; \; \; \; \; z_2 = x_2+y_2i
\\ z_1^* = x_1-y_1i \; \; \; \; \; z_2^* = x_2-y_2i
\\ | z_1 | = \sqrt{x_1^2+y_1^2} \; \; \; \; \; | z_2 | = \sqrt{x_2^2+y_2^2}
\\ z_1+z_2 = x_1+x_2 + (y_1+y_2)i
\\ \left | z_1+z_2 \right |^2 = \left ( \sqrt{(x_1+x_2)^2+(y_1+y_2)^2} \right ) ^2 = (x_1+x_2)^2 +(y_1+y_2)^2
\\ z_1z_2^* = x_1x_2 -x_1y_2i +y_1ix_2 -y_1iy_2i = x_1x_2+y_1y_2 +(x_2y_1-x_1y_2)i
\\ z_1^*z_2 = x_1x_2 +x_1y_2i -y_1ix_2 -y_1iy_2i = x_1x_2+y_1y_2 +(x_1y_2-x_2y_1)i
\\ z_1z_2^* + z_1^*z_2 = 2(x_1x_2+y_1y_2) = \text{2Re}(z_1z_2^*) = \text{2Re}(z_1^*z_2)
\\ |z_1|^2+|z_2|^2 + z_1z_2^* + z_1^*z_2 = x_1^2+y_1^2 + x_2^2 + y_2^2 + 2(x_1x_2+y_1y_2)
\\ = (x_1^2 + 2x_1x_2 + x_2^2) + (y_1^2 + 2y_1y_2 + y_2^2) = (x_1+x_2)^2 +(y_1+y_2)^2
\\
[/math]
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>>9160456
Yes, that's it.
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What do [math]| - \rangle[/math] and [math]\mid + \rangle[/math] denote?
My understanding is that brakets just denote the standard hermitian scalar product, no?
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>>9160481
Awesome.
Lastly it is asking me to find the probabilities of finding spin up or spin down along all THREE cartesian axes. Use braket notation for the whole thing.
I don't see what differences would be between the axes? I can do the calculation for the z-axis, but what would the difference be for the x/y axes?
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Bra-ket notation is such cancer.
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>>9160482
In this context, it's the spin state of a particle.
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>>9160427
Consider the complex conjugate to just mean that i becomes (-i)
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>>9160886
brilliant!
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>>9160901
why would this be brilliant? its the definition of complex conjugate.
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