Determine the direction of a spin state given the state

AI Thread Summary
The discussion focuses on determining the direction of a spin state using a normalized wave function. The initial state was modified to include an imaginary component, leading to a normalized state that allowed for the calculation of cosine and sine values related to the angles of the spin state. Participants noted that the coefficients of the states correspond to sine and cosine functions, which are crucial for determining the angles. Suggestions were made to express the wave function in terms of phase factors and to normalize it correctly, leading to a clearer representation of the spin state. The final conclusion emphasized the relationship between the coefficients and the angles in the Bloch sphere representation.
Foracle
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Homework Statement
Given an unnormalized spin state
##\Psi=(1+i)|+>-(1+i\sqrt{3})|->##
which direction does this spin point to?
Relevant Equations
##|n;+> = cos\frac{\theta}{2}|+>+sin\frac{\theta}{2}e^{i\phi}|->##
From the relevant equation above, there is not imaginary part in the |+> state, so I multiplied the state by (1-i). The state is then :
##\Psi=(2)|+>-(1+\sqrt{3})+i(\sqrt{3}-1)|->##
Then I normalize it :
##\Psi=(\frac{1}{\sqrt{3}})|+>-\frac{1}{2\sqrt{3}}(1+\sqrt{3})+i(\sqrt{3}-1)|->##

From the |n;+> equation above, I concluded from the |+> part that :
##cos\frac{\theta}{2}=\frac{1}{\sqrt{3}}##
But when I did the same thing from the |-> part, I got different value from ##cos\frac{\theta}{2}##

My guess is that this method of determining direction of spin state is probably wrong.

Edit: Somehow the Latex is not working, I'm trying to figure out how to fix this
 
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I think your method is good.
Notice that the coefficient of the - state gives you ##\sin{\frac{\theta}{2}}##, not the cosine, maybe that's the problem.
To use LaTeX you need to use the $ or the # symbols (2before and 2 after the expression)
Take a look at the LaTeX Guide
 
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Gaussian97 said:
I think your method is good.
Notice that the coefficient of the - state gives you ##\sin{\frac{\theta}{2}}##, not the cosine, maybe that's the problem.
To use LaTeX you need to use the $ or the # symbols (2before and 2 after the expression)
Take a look at the LaTeX Guide
Oh yeah, that's right. Thanks man! Can't believe I missed such that 😅.

About the Latex, I have added the # symbols but it seems like its still not working. Do you know what's wrong with it?
Edit: Never mind, the Latex works now. Thanks again!
 
Here's an idea: $$\tan \frac \theta 2 = \frac{|1 - i\sqrt 3|}{|1+i|} = \sqrt 2$$
 
Here is another idea. You have found that
##\Psi=(\frac{1}{\sqrt{3}})|+>-\frac{1}{2\sqrt{3}}(1+\sqrt{3})+i(\sqrt{3}-1)|->## from which you can identify

##\sin\dfrac{\theta}{2}e^{i\phi}=-\dfrac{1}{2\sqrt{3}}(1+\sqrt{3})+i(\sqrt{3}-1).##
Separate real and imaginary parts, $$\sin\frac{\theta}{2}\cos\phi=-\dfrac{1}{2\sqrt{3}}(1+\sqrt{3})~;~~\sin\frac{\theta}{2}\sin\phi=\sqrt{3}-1$$ and get ##\tan\phi## from this.
 
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I have the following suggestions. First, write the wave function in terms of phase factors. We have.
$$
(1+i)|+>= \sqrt{2}e^{i\frac{\pi}{4}}|+>
$$
$$
-(1+i\sqrt{3})|->=-2e^{i\frac{\pi}{3}}|->=2e^{i\frac{4\pi}{3}}|->
$$
$$
|\psi>=\sqrt{2}e^{i\frac{\pi}{4}}|+>+2e^{i\frac{4\pi}{3}}|->
$$
Now normalize the w.f. Since
$$
\sqrt{2}^2 + 2^2=6
$$
$$
|\psi>=\frac{1}{\sqrt{6}}(\sqrt{2}e^{i\frac{\pi}{4}}|+>+2e^{i\frac{4\pi}{3}}|->)
$$
$$
=\sqrt{\frac{1}{3}}e^{i\frac{\pi}{4}}|+>+\sqrt{\frac{2}{3}}e^{i\frac{4\pi}{3}}|->
$$
With the Bloch sphere representation we know that the quantum state doesn't change if we multiply the w.f. by any number of unit norm, i.e. ##e^{i\zeta}|\psi>=|\psi>##. For our w.f. multiply by ##\zeta=-i\frac{\pi}{4}## to get
$$
\sqrt{\frac{1}{3}}|+>+\sqrt{\frac{2}{3}}e^{i\frac{13\pi}{12}}|->
$$
Thus
$$
\cos(\frac{\theta}{2})=\sqrt{\frac{1}{3}}
$$
$$
\phi=\frac{13\pi}{12}
$$
 
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A neat suggestion.
 
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