Trying to understand Schrodinger's Equation

[hover]

Hello,

It's only been recently that I have acquired the math skills to deal with the time independent version of Schrödinger's Equation which is:

$$\frac{-\hbar^2}{2m} \frac{d^2}{dx^2}\Psi(x) + U(x)\Psi(x) = E\Psi(x)$$

I tried to derive a wavefunction that deals with a particle in a confined box with infinite walls as shown http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/pbox.html#c1"

I was able to get something similar to:

$$\Psi(x) = A*sin(kx)$$

What somewhat baffles me is how they define lambda( the wavelength within k as shown http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/schr.html#c2"). Perhaps I am missing the obvious but why would the biggest wavelength be
$$\lambda = 2L$$
Why not have
$$\lambda = 4L$$
Or
$$\lambda = 7L$$
What makes $$\lambda = 2L$$ the biggest wavelength value. This would clear up a ton.
The second and last question I have(so far) is what particular value of n does a particle need to take? What does the value of n depend upon?

Thanks!

 

[Black Integra]

$\frac{-\hbar^2}{2\m} \frac{d^2}{dx^2}\Psi(x) + U(x)\Psi(x) = E\Psi(x)$

fx'd

In case you want to know, I'll show you a full proof of this.

First, let's define a function for this infinite potential well

U(x)={∞; x<0 or x>L}={0; 0≤x≤L}

So the Schrödinger's equation for 0≤x≤L become:
(-ħ²/2m)ψ''=Eψ

We have
ψ'' + (2mE/ħ²)ψ = 0

The solution of an equation is
ψ=Acos(kx) + Bsin(kx)
where k=sqrt(2mE/ħ²) and A,B are constant.

from the condition: ψ(0)=0;
ψ(0) = Acos(0) + Bsin(0) = A = 0

Thus the equation become;
ψ=Bsin(kx)

also; ψ(L)=0;
ψ(L) = Bsin(kL) = 0
But for B≠0;
kL=n¶ or
L=n¶/k

but for k=2¶/λ

so
L = nλ/2

for the minimum state n; n=1
λ = 2L #

and that's the reason for this particular number.
guess it's clear :)

 

[jtbell]

What somewhat baffles me is how they define lambda( the wavelength within k as shown http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/schr.html#c2"). Perhaps I am missing the obvious but why would the biggest wavelength be
$$\lambda = 2L$$

It follows from these requirements:

1. The wave function must be zero outside the box, because of the "infinitely-high" walls.

2. The wave function must be continuous with no abrupt "jumps" from one point to the next.

This means that the equation you use for the wave function inside the box must evaluate to zero at the walls. What's the longest wavelength that gives you a node (zero point) at both walls?

 

[hover]

Thanks for the responses! I like both solutions that are proposed here. On one hand, Black Integra shows how the math just makes the λ=2L fall right out. On the other hand, jtbell shows how the λ=2L falls right out by stating 2 requirements. The longest wavelength must be λ=2L since the wave gives you a zero point starting at a the beginning of wavelength and if you add a half the wavelength you end up at zero again. Very nice solutions.

Just one more question. How does one determine what value of n the wave is in? Since n relates to the amount of energy, does n always take the lowest possible value?

http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/schr.html#c3"

$$E_n = \frac{n^2h^2}{8mL^2}$$

Also is E_n a kinetic energy or potential energy? I assume it is a kinetic energy since that is how it seems to be described in the link.

 

[Black Integra]

No problems, hover :)

If you still prefer equations for an explanation, here we go.

What is the value of E_n?
From the relation I've said
L=n¶/k ----> k=n¶/L
and
(kħ)²=2mE

So we have a Total Energy at state n

E_n=(1/2m)(n¶ħ/L)²=(1/8m)(nh/L)²
It's total energy, due to the variable E in Schrödinger's equation.
Anyway, the potential of system is zero at 0≤x≤L.
So, in this case, total energy is equal to kinatic energy.

What exactly is the value n?
n is considered to be a state number of each system.

For example, the value of n in the energy level of hydrogen atom.
In this system, i.e. infinite square well, there're many states depending on the value n(or total energy).
Each energy will give different wave function(different n).
At n=1, it will give the lowest energy. This energy is called 'zero-point energy'

The value n is a discrete value. So it could not be any value. It just like a resonance of LRC-circuit or spectrum of Hg-atom.