Newbie questions: wave and particle viewpoints

In summary, the author explains the limitations of the particle and wave viewpoints and how the squared absolute value of the probability amplitude gives a wave that travels in the +r direction. He also explains why t and r variables are not used and why the squared absolute value of a complex number is always real. Finally, he explains how to make graphs and shows how taking k out of ωt results in a wave equation that always lies between -1 and 1.
  • #1
Fyreth
8
0
I'm still in high school. I'm reading about physics as a hobby.

In http://student.fizika.org/~jsisko/Knjige/Opca%20Fizika/Feynman%20Lectures%20on%20Physics/Vol%203%20Ch%2002%20-%20Relation%20of%20Wave%20&%20Particle%20Viewpoints.pdf the limitations of the particle and wave viewpoints are described.

I've read that the probability of an event is the squared absolute value of the probability amplitude. Why would you do that? Wouldn't just taking the square work as well?

On the first page of that document there is this probability amplitude: e[itex]^{i(ωt-k \bullet r)}[/itex]
Why is kr substracted from ωt? Why aren't the phases added to each other?
It also says that the squared absolute value of this amplitude is constant but aren't t and r variables? Why would it be constant?
 
Physics news on Phys.org
  • #2
Fyreth said:
I've read that the probability of an event is the squared absolute value of the probability amplitude. Why would you do that? Wouldn't just taking the square work as well?

The probability amplitude is a complex number, in general. Squaring a complex number gives another complex number. Probability has to be a real number. The squared absolute value is always real.

On the first page of that document there is this probability amplitude: e[itex]^{i(ωt-k \bullet r)}[/itex]
Why is kr substracted from ωt?

It gives you a wave that travels in the +r direction.

Why aren't the phases added to each other?

They can be added, in which case the wave travels in the -r direction.

It also says that the squared absolute value of this amplitude is constant but aren't t and r variables? Why would it be constant?

To get the squared absolute value of a complex number, we multiply the number by its complex conjugate (replace i with -i and vice versa).

[tex]e^{i\theta}e^{-i\theta} = e^{i\theta-i\theta} = e^0 = 1[/tex]
 
Last edited:
  • #3
jtbell said:
It gives you a wave that travels in the +r direction.

They can be added, in which case the wave travels in the -r direction.

Why? Does a negative phase mean that the wave is traveling in a positive direction?
 
  • #4
I think what they explain is true. there are implicit physics characteristics of wave that represent by complex number.
for more understanding purpose, you can read Marry L.Boas, Mathematical Method for Physics. You will understand with what Chaucy approach about complex number
 
  • #5
Fyreth said:
Why? Does a negative phase mean that the wave is traveling in a positive direction?

To convince yourself of this, I think it's best to make some graphs. Pick numbers for [itex]\omega[/itex] and k, set t = 0, and draw a graph of y versus x for [itex]y = \cos (\omega t - kx) = \cos (-kx)[/itex]. Then set t = 0.1, say, and draw another graph. Then set t = 0.2 and do another one.

Repeat for [itex]y = \cos (\omega t + kx)[/itex].

You need to use the cosine (or sine) instead of [itex]e^{i(\omega t - kx)}[/itex] because it's kind of hard to draw graphs of complex numbers.
 
  • #6
Hi, I've just started a Masters in Acoustics after a ten year gap from education, so I've spent the last few months getting my head around this subject too.

One really important thing I realized recently is that ωt = kx. Here's the proof,

c=fλ, f=c/λ, λ=c/f
k=2π/λ,
c=x/t, x=ct, t=x/c
ω = 2πf
→ ω=2πc/λ → ω=ck → ω=kx/t → ωt = kx

So what's the point then? Shouldn't the whole thing just be cos(0)=1?

Engineering Noise Control: Theory and Practice is a good book for this. It gives this expression for the wave equation, φ=Acos(k(ct±x)+β

Here ± is used because a wave can go in either direction, but more importantly, k has been taken out, clearly showing the two most important things within the bracket - ct & x. What this does is allow us to consider the wave when time is zero but at a distance x, or when time is 20 ms and the distance is zero - or at some combination of the two.

k=2π/λ and this 2π makes the cos bracket give a solution that will always be between -1 and 1.

β is the phase - and is easiest considered as a value that shifts the entire wave left or right by a given amount. The phase was also a part of something that confused me for a long time, which was the addition of waves. Adding two incoherent sources together (two violinists as opposed to one) increases the SPL by 3 dB, but adding two coherent sources (two speakers as opposed to one - putting out exactly the same signal which is in phase) increases the SPL by 6 dB. When two signals are in phase with each other you get an extra 3 dB increase.

There isn't actually a standard convention for using positive or negative in this respect (Cox, 2009, s. 1.4.1).
 

What is the wave-particle duality?

The wave-particle duality is a fundamental principle in quantum mechanics that states that particles can exhibit both wave-like and particle-like behavior. This means that they can behave like waves in some experiments and like particles in others.

Can light be both a wave and a particle?

Yes, light is a perfect example of the wave-particle duality. In some experiments, it behaves like a wave, such as when it diffracts or interferes. In others, it behaves like a particle, such as when it is emitted or absorbed by matter.

What is the difference between the wave and particle viewpoints?

The wave and particle viewpoints are two different ways of describing the behavior of particles. In the wave viewpoint, particles are described as continuous waves, while in the particle viewpoint, they are described as discrete particles. These two viewpoints are both valid and complementary, and they are necessary to fully understand the behavior of particles.

Which viewpoint is correct?

Neither the wave nor the particle viewpoint is correct or incorrect. They are both just different ways of describing the same phenomenon. In some experiments, one viewpoint may be more useful than the other, but in general, both viewpoints are necessary to fully explain the behavior of particles.

How does the wave-particle duality affect our understanding of the universe?

The wave-particle duality has revolutionized our understanding of the universe, particularly in the field of quantum mechanics. It has led to breakthroughs in technology, such as the development of lasers and transistors. It has also challenged our traditional understanding of particles and their behavior, leading to new theories and interpretations of the universe.

Similar threads

Replies
17
Views
1K
Replies
1
Views
4K
  • Mechanics
Replies
4
Views
14K
Replies
1
Views
2K
  • General Discussion
Replies
4
Views
605
  • High Energy, Nuclear, Particle Physics
Replies
1
Views
2K
Replies
1
Views
6K
  • Introductory Physics Homework Help
Replies
4
Views
3K
  • High Energy, Nuclear, Particle Physics
Replies
2
Views
2K
Back
Top