Understanding the de Broglie Wave Function - Need a Little Help?

[Roodles01]

Just starting QM & am looking at TISE & how it builds from start & need a little help understanding.

OK, for a free particle the de Broglie wave function is

ψdB(x,t) = Ae^i(kx-ωt)
where A is a complex constant

this corresponds t a free particle with momentum magnitude
p = h/λ = hbar k

& energy
E = hf = hbar ω

good so far . . . . .

my textbook then asks what partial differential equation this wave fuction satisfies & suggests a suitable pde;

i hbar dψ_dB/dt = - hbar²/2m d²ψ_dB/dx²


Really not sure why this partial differential equation. What goes on?
Help if possible, please.

 

[bapowell]

That's the Schroedinger equation for the free particle. Although it's not a traditional wave equation (it's only first order in time) it does have wave solutions (e.g. the de Broglie function that you've written down). Can you see how that happens?

The derivation of this equation is a little murky, but it is essentially based on conservation of energy, linearity, and the fact that the evolution of the quantum state is an eigenvalue problem.

 

[Roodles01]

OK I have a worked example where it verifies the deBroglie wave function

ψdB(x,t) = Aei(kx-ωt)

provided that ω & k obey the condition hbarω = (hbar k)2/2m.

It goes on to find the Schrödinger equation fr a free particle

i hbar dψdB/dt = - hbar2/2m d2ψdB/dx2

which is great.

So is the pde I asked about earlier just a something I don't need to know how to derive or how it came about, but just IS.

 

[bapowell]

Hard to say. The traditional way of teaching quantum mechanics is to simply take the SE at face value. Personally, I think that's fine for students just getting started with the concepts. It's probably good to have in your mind that physically, the SE is essentially an expression of conservation of energy; it's first order in time to avoid negative energy solutions that occur with the traditional wave equation. Where it gets strange is the use of operators in place of the previously classical quantities of momentum and energy. I'm not aware of a rigorous derivation of this fact from any more basic principle, but others might be able to chime in here. Generally, the formulation of quantum mechanics as an eigenvalue problem -- with observables defined as the eigenvalues of Hermitian operators -- is taken to be an axiom.

 

[jtbell]

provided that ω & k obey the condition hbarω = (hbar k)2/2m.

Note (if you haven't already) that this is just E = p²/2m, the relationship between kinetic energy and momentum, which you can derive from E = mv²/2 and p = mv. So this condition is justifiable for non-relativistic particles.

For relativistic particles, we have to take a different route which leads us to the Dirac equation.