Exercise about the wavefunction

In summary: You should be able to see the correct pattern from that.In summary, the conversation discusses finding the wavefunction and distance between two points on an ideal rope with a wave moving at velocity 20 m/s. The wavefunction is given by ##\xi(x,t)=0.1 \mathrm{sin}(6t-0.3x )## and the distance between two points at a certain time is given by ##\frac{\pi-2\arcsin(0.2)}{0.3}##. The mistake in the derivation was not considering points where ##\xi = -0.02## m and a sketch can help visualize the correct pattern.
  • #1
Soren4
128
2

Homework Statement


[/B]
Consider an ideal rope where there is a wave moving at velocity ##v=20 m/s##. The displacement of one end of the rope is given by
$$s(t)=0.1 \mathrm{sin}(6 t)$$
a) Find the wavefunction ##\xi(x,t)##, knowing that it is progressive
b) Find the distance ##\delta## (in absolute value) between two points of the rope, that, at a certain time istant, are displaced from Equilibrium position of ##0.02 \mathrm{m}##
##[\mathrm{Result} \, \delta=1.34 m]##

Homework Equations


[/B]
A general wavefunction is
$$\xi(x,t)=\xi_0 \mathrm{sin}( k x-\omega t +\psi)\tag{1}$$

The Attempt at a Solution


[/B]
a) My boundary condition for ##(1)## is that
$$\xi(0,t)=0.1 \mathrm{sin}(6 t)\tag{2}$$
Now is it correct to conclude that ##\xi_0=-0.1 m##, ##\omega=6 rad/s## and ##\psi=0##?
If so, then, considering also ##k=\frac{2 \pi}{\lambda}=\frac{\omega}{v}=0.3 \frac{1}{m}##
$$\xi(x,t)=0.1 \mathrm{sin}(6t-0.3x )\tag{3}$$

b)Here is the problem. I would say that

$$0.02=0.1 \mathrm{sin}(6t-0.3x )\implies (6t-0.3x)=\arcsin(0.2)+2n\pi \vee \pi-\arcsin(0.2)+2n\pi \implies |x_2-x_1|=\frac{\pi-2\arcsin(0.2)}{0.3}$$

But this does not give the correct result.

Where did I go wrong in this problem? Any suggestion is highly apprectiated.
 
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  • #2
Apparently they want the smallest Δx between two points for which the absolute value of the displacement ##\xi## is 0.02 m.
 
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Likes Soren4
  • #3
Thanks for the reply!

Yes that's part of the question I guess, could you suggest where I went wrong in my derivation? I still do not see the mistake..
 
  • #4
It looks like your solution only considers values of ##x## where ##\xi = +0.02## m. What about points where ##\xi = -0.02## m? It helps to make a sketch of the sine wave at some instant of time and mark the points on the x-axis where ##\xi = \pm 0.02## m.
 

What is the wavefunction?

The wavefunction is a mathematical function used in quantum mechanics to describe the quantum state of a particle or system of particles. It contains information about the position, momentum, and other properties of the particles.

What does the wavefunction represent?

The wavefunction represents the probability amplitude of the particle or system being in a particular state. It is not a physical quantity, but rather a mathematical tool used to make predictions about the behavior of quantum systems.

How is the wavefunction related to the Schrödinger equation?

The Schrödinger equation is a fundamental equation in quantum mechanics that describes how the wavefunction evolves over time. It is used to calculate the future state of a quantum system based on its current state and the potential energy of the system.

Can the wavefunction be directly measured?

No, the wavefunction is a mathematical concept and cannot be directly measured. However, the predictions made using the wavefunction can be tested and verified through experiments.

What are the units of the wavefunction?

The wavefunction does not have any physical units as it is a mathematical function. However, the square of the wavefunction has units of probability, typically measured in square meters or inverse seconds.

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