Linked Questions

3
votes
2answers
1k views

Using complex numbers to represent waves [duplicate]

When talking about a plane wave of the form $$\vec E=\vec E_0\cos(\vec k \cdot \vec r-\omega t)$$ We can replace it by $$\vec E=\vec E_0\exp[i(\vec k \cdot \vec r-\omega t)]$$ so that it is easier for ...
3
votes
2answers
348 views

Using complex exponential to represent waves in EM [duplicate]

Ever since we've been using exponentials to work with electromagnetic waves, I've been confused about the imaginary portion and want to confirm my thinking. What does the imaginary portion represent? ...
0
votes
1answer
48 views

Step in the derivation of complex wave notation [duplicate]

I'm reading Hecht's Optics and I have a problem understanding a step in the derivation of the complex notation of waves He writes that the wave equation for a harmonic wave can be written as $\Psi(x,...
0
votes
2answers
51 views

Michelson interferometr electromagnetic wave formula [duplicate]

I don't understand why we use this formula: Instead of this formula: I mean, why we ignore $\omega\cdot t$ part.
-1
votes
1answer
44 views

Problems with function transforming [duplicate]

So in quantum mechanics, most of the people uses exponential equation to handle the problem and present the relationship. Another word, we often handle the classical wave function in trigonometric way ...
4
votes
3answers
1k views

What is the physical significance of $i=\sqrt{-1}$? [closed]

What is the physical significance of $i=\sqrt{-1}$?
3
votes
3answers
426 views

How can $F_0\cos\omega t$ change to $F_0e^{i\omega t}$ in driven oscillator equation?

I have one thing that confuses me on deriving the solution for the Linear Forced Oscillator. Suppose we have the equation as $$ma + rv + kx = F_0 \cos \omega t$$ What confuses me is when the driving ...
3
votes
1answer
1k views

What is the advantage of using exponential function over trigonometric function in analyzing waves?

A.P.French in his book Vibrations and Waves writes: . . . Why should the exponential function be such an important contribution to the analysis of vibrations? The prime reason is the special ...
3
votes
3answers
207 views

Is plane wave equation $\Psi(\mathbf r,t)=\Psi_0e^{i(\mathbf k \cdot \mathbf r-\omega t)}$ for quantum-mechanical wavefunctions really complex? [duplicate]

The equation given for plane sine wave (for instance used to "derive" the Schrödinger equation) is $$ \Psi(\mathbf r,t)=\Psi_0e^{i(\mathbf k \cdot \mathbf r-\omega t)} $$ I would have assumed that ...
3
votes
1answer
997 views

Meaning of imaginary part of complex field amplitude for waveguide modes (e.g. TE, TM, HE, EH)

In classical waveguide analysis (e.g. for optical fibers, as in the notes Modal analysis of step-index fibers, ECE 4006/5166 Guided Wave Optics, Robert R. McLeod, University of Colorado), one can find ...
0
votes
6answers
226 views

How can $e^{i\theta}$ ever be a real quantity?

From Euler's formula, $e^{i\theta} = \cos(\theta) + i\sin(\theta)$, which seems to be a complex quantity involving real and imaginary parts. Yet, D.J. Griffiths in his book on electrodynamics mentions ...
2
votes
1answer
286 views

Complex-valued current density?

I'm reading in several places (e.g., here), that the current density is supposed to be in the form $\mathbf{j}=j(r, z) \exp(i\omega t) \,\mathbf{e}_\varphi, [...]$ This implies that the current ...
1
vote
2answers
111 views

Can we observe light oscillations with an ultra fast sensor?

In classical wave optics, if human sensor is able to detect amplitude changes at a frequency of visible light (i.e. at $10^{14}$ Hz order), for wave $$u(x,z) = A(x) e^{j (k_1 x + k_2 z + \omega t)}$$ ...
0
votes
1answer
157 views

Why consider only real part when summing several simple harmonic motions?

I have been studying vibrations and I stumbled upon the overlapping of simple harmonic motions. Consider the case where the number of oscillators $n$ is $n \gg 1$, all of them have the same angular ...
0
votes
3answers
207 views

Understanding reflection through polarization

I came across the following explanation: The source of this so-called reflected light is not simply that the incident beam is reflected; our deeper understanding of this phenomenon tells us that ...

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