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Important Note: For "layman" read "next to zero understanding of QM mathematics"

I am reading Quantum Mechanics: The Theoretical Minimum. In Chapter 2 on Quantum States, the following is presented:

$$|A\rangle = a_u |u\rangle + a_d |d\rangle$$

and

$$a_u = \langle u|A\rangle$$ $$a_d = \langle d|A\rangle$$

where $|A\rangle$ is a generic state that can represent the state of a spin, and $a_u$, $a_d$ are components of $|A\rangle$ along the basis vectors $|u\rangle$ and $|d\rangle$.

The book then states that the probability that the spin will be up if measured along the z axis is $a_u^* a_u$, i.e., the square of the magnitude of the probability amplitude.

But then it presents:

$$P_u = \langle A|u\rangle \langle u|A\rangle$$

where $P_u$ is the probability of the spin being measured as up.

My Question:

Why isn't

$$P_u = \langle u|A\rangle \langle u|A\rangle$$

given what was stated earlier?

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  • $\begingroup$ To format equations, use LaTex as in an usual .tex file. The complex conjugation of $\langle u, A\rangle$ is $\langle A,u\rangle=(\langle u, A\rangle)^*$. $\endgroup$ – yuggib Feb 21 '16 at 13:09
  • $\begingroup$ @yuggib: Apologies, I'm not sure I understand your contribution. Is there a Latex app for android and if so, how would I add the .tex file (which I assume is generated from the Android app, if it exists) to my question? $\endgroup$ – MonaLisaOverdrive Feb 21 '16 at 13:18
  • $\begingroup$ No, he meant to say that you can type math as you would in a tex-file on this site, using $ [typeset your math here] $ $\endgroup$ – Danu Feb 21 '16 at 14:04
  • $\begingroup$ They've used MathJax, which is enabled on this site. This Tutorial explains some of what they mean. $\endgroup$ – Kyle Kanos Feb 21 '16 at 18:56
  • $\begingroup$ Thanks all. And thank you to whomever fixed the equations in my original post $\endgroup$ – MonaLisaOverdrive Feb 21 '16 at 21:06
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The coefficient $a_u = \langle u|A\rangle$ is allowed to be a complex number. The square of a complex number is also complex in general, but we do not want our probability to be complex. We therefore multiply $a_u$ with its complex conjugate $a_u^* = \langle A|u\rangle$, which always gives a real number.

Suppose that $a_u = x + yi$. The complex conjugate is given by $a_u^* = x - yi$. We can calculate that $$a_ua_u=(x+yi)^2 = x^2 - y^2 +2xyi.$$ This number has an imaginary part; it is not a real number. On the other hand, $$a_ua_u^*=(x + yi)(x - yi) = x^2 + y^2,$$ which is indeed a real number.

If you interpret the complex number $a_u$ to be vector/arrow in the complex plane, then $a_ua_u^*$ is the square of the length of the vector. So, besides being garanteed real, the product $a_ua_u^*$ also makes sense from a geometric point of view.

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  • $\begingroup$ Thank you. The book does not mention that the probability is equal to the probability amplitude multiplied by its complex conjugate, it just says that it is equal to the probability amplitude squared. It also writes $a_u*a_u$…which looks like $a_u x a_u$ Odd that MathJax isn't allowed in comments… $\endgroup$ – MonaLisaOverdrive Feb 21 '16 at 21:12
  • $\begingroup$ The magnitude or "size" of the probability amplitude $a_u$ -- or any other complex number -- is defined to be $|a_u| = \sqrt{a_ua_u^*}$. Squaring this magnitude gives us the desired result $|a_u|^2 = a_ua_u^*$. If the probability amplitude happens to be real, then $a_ua_u^*=a_u^2$, because the complex conjugate of real number is the number itself $\endgroup$ – vosov Feb 21 '16 at 22:44
  • $\begingroup$ I don't understand the comment…are you explaining more about your answer? $\endgroup$ – MonaLisaOverdrive Feb 23 '16 at 17:43
  • $\begingroup$ I am sorry if I am confusing you. I don't have your book at hand and am trying to give you some pointers as to why the book tells you that simply squaring the amplitudes is ok. The most likely reason is that the examples that you are looking at are using real coefficients $a_u$, rather than complex ones. If that is the case, you can simply square the coefficients to calculate the probability. When the coefficient is complex, you must multiply with the complex conjugate to get the correct probability. $\endgroup$ – vosov Feb 24 '16 at 16:37
  • $\begingroup$ No worries. To clear some things up…1) $a_u$ is a complex number in the book. 2) the book says that the probability amplitude must be squared in order to get the probability BUT it does NOT say that if the probability amplitude is a complex number, multiply it against its complex conjugate to get the probability $\endgroup$ – MonaLisaOverdrive Feb 24 '16 at 20:08

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