Quantum Mechanics Proof Homework Help

[SlushmanIU]

I was asked to prove that any function u(z,t)=f(z-vt) is a solution of the wave equation
∂2u/ dt2= v2 · ∂2u/dz2

I know that v is constant and z and t are independent. I've tried looking at Lorentz law but I am getting nowhere fast. Please help!

 

[BvU]

Hello Slushman, and welcome to the wonderful world of PF :)

We don't have many rules (just a bunch of well-meant guidelines, which please read). They do require (so you could construe that as a rule) some effort on your part in the sense that you show your attempt at solution. They also want you (so you could construe that as a rule as well -- but it's all well meant!) to use the template, which happened to disappear as if by magic from your post. Pity, I could have helped immediately, instead of tomorrow morning (it's late here, but perhaps others ...)

1. Homework Statement

2. Homework Equations

3. The Attempt at a Solution

When I fill in the propposed solution, I get ...​

 

[Goddar]

Hi. This is pure mathematics so Lorentz has nothing to do with your proof:
How do you take the partial time-derivative of f(z-vt)? That is: f[z(t) -vt]?
Look up a calculus book if that's unfamiliar..

 

[BvU]

Slushman still there ?

 

[paranormal]

I understand your frustration with trying to prove this concept, but let me assure you that it is possible. First, let's start by breaking down the wave equation you are given:

∂2u/ dt2= v2 · ∂2u/dz2

This equation represents the second derivative of u with respect to time (t) equal to the product of the constant v squared and the second derivative of u with respect to z. This is known as the wave equation and is commonly used to describe wave-like phenomena in physics.

Now, let's look at the function u(z,t)=f(z-vt) that you are asked to prove is a solution of this wave equation. This function represents a wave traveling in the z-direction with a constant velocity of v. The value of u at any point (z,t) is equal to the value of the function f at the point (z-vt). This means that as time passes, the wave will move along the z-axis with a constant velocity of v.

To prove that this function is a solution of the wave equation, we need to show that it satisfies the equation for all values of z and t. Let's start by taking the first derivative of u with respect to t:

∂u/∂t = -vf'(z-vt)

We can see that the second derivative of u with respect to t is equal to:

∂2u/∂t2 = -vf''(z-vt)

Next, let's take the second derivative of u with respect to z:

∂2u/∂z2 = f''(z-vt)

Now, if we substitute these values into the wave equation, we get:

- vf''(z-vt) = v2f''(z-vt)

We can see that the v2 term cancels out, leaving us with:

- f''(z-vt) = f''(z-vt)

This shows that the function u(z,t)=f(z-vt) satisfies the wave equation, proving that it is a solution. This makes sense intuitively, as the function represents a wave traveling at a constant velocity in the z-direction, which is exactly what the wave equation describes.

In conclusion, by taking the second derivatives of the function u(z,t)=f(z-vt) and substituting them into the wave equation, we can see that it satisfies the equation for all