The probability as an absolut value of the square of the amplitude

[TubbaBlubba]

All right, I don't have a problme with the concept, just a specific question.

Is the absolute value of the amplitude abs(r^2 + (xi)^2) or abs(r^2) + abs((xi)^2)

Or, to put it in a simpler way - Do you absolute the value of the square of the imaginary part?

The difference would be, say

2^2 + (5i)^2 = 4 + (-25) = (-21)
abs(-21) = 21

and

2^2 + (5i)^2 = 4 + (-25)
abs(4) + abs(-25) = 29

The latter seems more physically sound to me, but the former seems more mathemathically sound. Can anyone clear this up for me?

 

[betel]

The imaginary part is the part standing next to the "i", so in your example the 5 itself. Then you get the absolute value of the complex number by
$$|z|^2 = \Re(z)^2+\Im(z)^2$$
Or by using the complex conjugate
$$|z|^2 = z \cdot \bar z$$

For the first part I suspect you write your complex number in polar coordinates
$$z = r \exp{i \xi}$$
In this case the absolute value would be just $$|z|=r$$

 

[TubbaBlubba]

Ah, I think I see, the absolute value is the distance from the origin? Thanks for clearing it up.

 

[betel]

Ah, I think I see, the absolute value is the distance from the origin? Thanks for clearing it up.

Yes. If you draw the complex number in the complex plane you can recognize the above formula as an application of Pythagoras' theorem.

 

[CellCoree]

I would say that both approaches are mathematically sound, but the interpretation and application may differ. Let's break it down further:

The square of the amplitude, represented as r^2, is a real number. Taking the absolute value of a real number simply removes any negative signs, resulting in a positive value. So, abs(r^2) would be the same as r^2.

The imaginary part, represented as xi or (xi)^2, is a complex number. Taking the absolute value of a complex number is a bit more nuanced. In the first approach, abs(r^2 + (xi)^2), we are taking the absolute value of the sum of two complex numbers, which results in the absolute value of the resulting complex number. In the second approach, abs(r^2) + abs((xi)^2), we are taking the absolute value of each individual complex number and then adding them together. Both approaches are valid, but they may have different interpretations and applications in different contexts.

In physics and engineering, the absolute value of the square of the amplitude may be used to calculate the intensity of a wave or signal, where the imaginary component represents the phase. In this case, the first approach may be more appropriate as it takes into account both the magnitude and phase of the signal. However, in pure mathematics, the second approach may be more commonly used as it simplifies the calculation by separating the real and imaginary components.

In conclusion, both approaches are mathematically sound, but the interpretation and application may vary depending on the context. It is important to understand the underlying principles and choose the appropriate approach for your specific problem.