Commutative Multiplication of Complex number magnitudes?

[IdealFugacity]

Hi, I'm studying physical chemistry (senior year of chem engineering), which is, as far as this post concerns, an intro to quantum mechanics. Currently working through section of the textbook on representing wave functions as complex functions.

As a brief exercise, there is an example problem covering finding the magnitude of some complex numbers (magnitude |f| = sqrt (f* f)...multiplying all i's by negative 1 for the f* term, if that serves as a reminder if necessary

so for example, the magnitude of (1+i) (sqrt2 + 5i) is found by finding:
sqrt[ (1+i)(sqrt2 + 5i)(1-i)(sqrt2 - 5i) ] = 3sqrt6

the other 3 have forms such as (1+sqrt3 i)/ (11-2i) , or
exp(sqrt2 i pi) * exp( -3 i pi) /
4exp (i pi/4)

For all of these, I found the correct answer the long way (sqrt f*f)
however, i noticed that if i find the magnitude of each individual term and pretending that this property was commutive (commutative?), i got the same answers, eg for the first example:
(1+i)(sqrt2 + 5i), multiplying the magnitude of the first term (sqrt 1sq plus 1sq = sqrt2), by the magnitude of the 2nd term (sqrt (2 plus 25) = sqrt27 = 3sqrt3, i got 3sqrt6 again. This worked for all 4 problems given in the book.

Is this dangerous, and I was lucky? Or will this property be true for future, more complicated, problems of this type?

Thanks for reading

[bunny]

EDIT for wrongness

[thylacine]

It is always true for complex numbers that the magnitude of the product is the product of the magnitudes.

For more info, check a high school maths textbook.

[jay_shark]

|a-bi|= sqrt(a^2+(-b)^2)

(a+bi)(c+di)= ac+ (a+d)i +(b+c)i+bdi^2 ; i^2=-1
=ac +(a+b+c+d)i -bd
=(ac-bd)+ (a+b+c+d)i

The magnitude of this vector is
sqrt[(ac-bd)^2 + (a+b+c+d)^2]

[IdealFugacity]

Thank you for the replies, but bunny, why did you retract your answer (identical to thylacine's, if i recall correctly)?

[bunny]

I just posted a "yes" answer. That's a pet hate of mine and since I didnt have time to post the full reason I took it back.

[flipdeadshot22]

Once you learn how do deal with complex numbers as geometric constructs, many of their funny properties become obvious. I exclusively visualize them via Euler's relation to show how they rotate in the complex-real plane:

E^(ix) = cos(x) + isin(x),

and you should get used to this form since it will be ubiquitous throughout your studies in quantum. In fact, the first wavefunction you will deal with, the free particle solution to the schrodinger equation, will be of this form, since it's nothing more than a plane wave.

[IdealFugacity]

[ QUOTE ]
Once you learn how do deal with complex numbers as geometric constructs, many of their funny properties become obvious. I exclusively visualize them via Euler's relation to show how they rotate in the complex-real plane:

E^(ix) = cos(x) + isin(x),

and you should get used to this form since it will be ubiquitous throughout your studies in quantum. In fact, the first wavefunction you will deal with, the free particle solution to the schrodinger equation, will be of this form, since it's nothing more than a plane wave.

[/ QUOTE ]

I understand (and am pretty sure I understood as of the last day or so) that, I'm not sure the point you are making as it relates to my original question.

Professor said he was pretty sure it was correct, the guy's kind of a ditz though

[flipdeadshot22]

[ QUOTE ]
[ QUOTE ]
Once you learn how do deal with complex numbers as geometric constructs, many of their funny properties become obvious. I exclusively visualize them via Euler's relation to show how they rotate in the complex-real plane:

E^(ix) = cos(x) + isin(x),

and you should get used to this form since it will be ubiquitous throughout your studies in quantum. In fact, the first wavefunction you will deal with, the free particle solution to the schrodinger equation, will be of this form, since it's nothing more than a plane wave.

[/ QUOTE ]

I understand (and am pretty sure I understood as of the last day or so) that, I'm not sure the point you are making as it relates to my original question.

Professor said he was pretty sure it was correct, the guy's kind of a ditz though

[/ QUOTE ]

It actually does relate in the way you can interpret a complex number as a sort of "vector", with an associated magnitude (don't take that too literally though) undergoing a rotation in the real-imaginary plane. I was just trying to show a different way of thinking about the same object that you are studying. This wiki article discusses it with much more clarity than I can:

http://en.wikipedia.org/wiki/Euler's_formula

It really is an extraordinarily powerful equation once you come to grips with it.

[jason1990]

The connection comes from the fact that a complex number can be written in the form z=re^{it}, where r=|z| and t is the angle in the complex plane between the positive real axis and the ray starting at the origin and passing through z. If w=qe^{is} is another complex number, then

zw = rqe^{i(t + s)}.

From here, it is clear that |zw|=rq=|z||w|.

[PairTheBoard]

Your observation is correct via De Moivre's Theorum. Any complex number can be expressed as

r[cos(t) + i*sin(t)]

where t is the angle with the x-axis and r is the magnitude of the complex number viewed as a vector on the plane. De Moivre's Theorum then says that for two complex numbers with magnitudes r1, r2 and angles t1,t2, the complex product of the numbers is:

r1[cos(t1) + i*sin(t1)] * r2[cos(t2) + i*sin(t2)] =

= r1r2[cos(t1+t2) + i*sin(t1+t2)]

You can prove this for yourself by multiplying the forms out and applying a couple of standard identities from trigonometry.

You can also see this working more easily if you accept the form given above for a complex number:

r[cos(t) + i*sin(t)] = r*e^(i*t)

and apply the law of exponenents. Although it's not so intuitive why that form makes sense.

PairTheBoard

[IdealFugacity]

Thanks Jason and PairTheBoard, those are excellent replies. Almost want to forward them to Dr. "Uh..Yeah, I think so." < - also tried to claim normal human body temp is 22 celsius

This course is making me wish I had chosen physics way back when, unfortunately my high school teacher wasn't really anything special, and the first course i had here was probably the closest thing to hell i experience. but this QM stuff is really piqueing my interest (not to mention my dad got his PhD in particle physics, so i guess it's in my blood)

Oh well, hopefully QM adoration is a phase I get out of before i'm in industry 6 months from now, hating my life because i wish i had done something else /images/graemlins/tongue.gif

[MusashiStyle]

[ QUOTE ]

Oh well, hopefully QM adoration is a phase I get out of before i'm in industry 6 months from now, hating my life because i wish i had done something else /images/graemlins/tongue.gif

[/ QUOTE ]

I love QM, but it's really no picnic to learn. I got an A in it in Undergrad and then I though I was some sort of theoretical whiz. I ended up dropping out of Grad QM because I couldn't handle it. Just so you know, it keeps going very deep/ fascinating subject but a hell of a lot of work.

[IdealFugacity]

[ QUOTE ]
[ QUOTE ]

Oh well, hopefully QM adoration is a phase I get out of before i'm in industry 6 months from now, hating my life because i wish i had done something else /images/graemlins/tongue.gif

[/ QUOTE ]

I love QM, but it's really no picnic to learn. I got an A in it in Undergrad and then I though I was some sort of theoretical whiz. I ended up dropping out of Grad QM because I couldn't handle it. Just so you know, it keeps going very deep/ fascinating subject but a hell of a lot of work.

[/ QUOTE ]

Oh, I by no means think I'm a whiz. I'm only studying the text so much because the non-book homework is so difficult and i'm productively procrastinating, and in addition diff eqs are probably the most difficult part of calculus for me (now anyway, weren't when i had the course, but my memory sucks). I'm making these posts from the point of view of observer who is enjoying the glimpses i am getting, not someone who thinks i'm gonna kick QM butt (though kicking QM butt relative to the rest of the class would be fun)