Quick complex numbers question in QM (probability amplitues)

[phil ess]

[RESOLVED] Quick complex numbers question in QM (probability amplitues)

Im a little confused here. I am reading in my textbook about probability amplitudes in Stern Gerlach measurements, and it says this:

We find the resulting probabilities for deflection of $$\left(\stackrel{\alpha}{\beta}\right)$$ in the x and y directions as:

$$Prob(\pm in x) = \left|\frac{\alpha\pm\beta}{\sqrt{2}}\right|^2 = \frac{1}{2}(|\alpha|^2 + |\beta|^2 \pm \alpha*\beta \pm \beta*\alpha) = \frac{1}{2} \pm Re(\alpha*\beta)$$

$$Prob(\pm in y) = \left|\frac{\alpha\mp i\beta}{\sqrt{2}}\right|^2 = \frac{1}{2}(|\alpha|^2 + |\beta|^2 \mp i\alpha*\beta \mp i\beta*\alpha) = \frac{1}{2} \pm Im(\alpha*\beta)$$


Now I understand where the amplitudes come from and everything, the thing I don't understand is the functions Re and Im, and how they simplified the amplitudes to

$$\frac{1}{2} \pm Re(\alpha*\beta)$$

and

$$\frac{1}{2} \pm Im(\alpha*\beta)$$

So can someone please explain these functions Re and I am and how they work here?

 

[Fredrik]

Any complex number z can be written in the form z=x+iy, where x and y are real numbers called the real part and the imaginary part of z respectively. x and y are uniquely determined by this decomposition, i.e. if x'+iy'=x+iy, then x=x' and y=y'. So there exists a function that takes a complex number to its real part, and a function that takes a complex number to its imaginary part. These functions are called Re and I am respectively.

Note that

z=Re z + i I am z
z*=Re z-i I am z

and that this implies

Re z=(z+z*)/2
Im z=(z-z*)/(2i)

Now what do you get when you compute the real and imaginary parts of $\alpha^*\beta$?

 

[phil ess]

OH! so:

1/2 + Re(a*b) = 1/2 + (a*b + b*a)/2 = 1/2 (1 + a*b + b*a)

but 1 = |a|² + |b|²

so we get:

1/2 + Re(a*b) = 1/2 (|a|² + |b|² + a*b + b*a)

woohoo, thanks a lot!

EDIT: Ok I am not seeing the I am part though:

1/2 + Im(a*b) = 1/2 + (a*b - b*a)/2i = (i + a*b - b*a)/2i ??

I can't seem to get the result in the book of 1/2 (|a|² + |b|² - ia*b + ib*a)

EDIT: Oh you just multiply top and bottom by i and it works out! got it, thanks agian!