Solutions of TISE - is it all about groping in the dark?

[DeShark]

Hi all, just joined the forums today, essentially to ask this one question which has been bugging me since I first started learning QM almost 2 years ago.

I'm basically trying to find solutions to the 1-D TISE for a square well potential.

My question is :- without making references to classical mechanics is it actually possible to predict anything using Quantum Mechanics and the schroedinger equation? Every book I've ever seen try to answer this problem has started with the line "We search solutions to the TISE in the form of standing waves". Is there anything to justify this action? Other than in the classical limit, this is how it works? This argument seems on thin ice...

I've heard people say that the Schroedinger equation is just like Newton's laws, but for quantum physics. Newton's laws don't ask me to search in the dark for solutions to F=ma. They ask me what F is and then tell me a. The schroedinger equation asks me what $$\hat{H}$$ is and then I have to prod blindly in the dark for my eigenfunctions... Or have I misunderstood?

Or is it more that once I know the wave function, the TDSE will tell me how it advances through time. i.e. is knowing the wavefunction in QM just like knowing the mass in Newton's mechanics and the TDSE is just like Newton's laws?

If so, is there any other way of working out the wavefunction than this crazy "let's guess that it's like this" mathematics? (I guess measurement is the only way, but then, we get collapse of the wavefunction sometimes... and can never measure everything that's contained in the wavefunction as a result)

Please help me, this is bugging me immensely. Without this understanding, I don't think I can take QM seriously.

 

[jtbell]

What you're basically asking is, "how do I solve a differential equation systematically?" The one-dimensional TISE is a linear second-order differential equation. There are general methods for solving linear differential equations, which are covered in undergraduate textbooks on differential equations.

 

[DeShark]

What you're basically asking is, "how do I solve a differential equation rigorously?" The one-dimensional TISE is a linear second-order differential equation. There are general methods for solving linear differential equations, which are covered in undergraduate textbooks on differential equations.

Hi, this response helped more than I thought it did upon first reading. I always knew that it was the solution to a linear second order ODE, but it was only when I read into the subject that I properly understood the importance of the uniqueness theorem. I'm still having difficulty getting my head round it, but from what I understand, there's *only one solution* to the TISE. It seems almost impossible that there can only be one solution to a differential equation. I mean, there's an infinite amount of functions out there and only one of these fits the bill, even though it can be expressed in numerous ways... Is this right? That's pretty awesome... Thanks a lot! This really helped my understanding.

 

[jtbell]

but from what I understand, there's *only one solution* to the TISE.

The uniqueness theorem applies to general solutions, before applying boundary conditions, I think. (For the infinite square well, the boundary conditions are that $\Psi$ must be zero at the walls.) For a second-order differential equation there is a single general solution, with two arbitrary constants. Conceptually, you can think of solving the DE as equivalent to integrating the second derivative twice, and each integration gives you an arbitrary constant.

Inside the well, there are actually two common ways of writing the general solution:

$$\Psi = A \cos (kx) + B \sin (kx)$$

$$\Psi = C e^{ikx} + D e^{-ikx}$$

where A and B are the arbitrary constants for the first one, and C and D for the second. However, these are really the same solution, because you can rewrite one into the other using identities like

$$e^{ikx} = \cos (kx) + i \sin (kx)$$

Then, after you've gotten the general solution, you apply the boundary conditions, and get the values of the arbitrary constants which work for your situation (infinite square well). These are the particular solutions for the infinite square well, which give you the various energy levels (energy eigenstates).

 

[George Jones]

Also, solving F = ma involves a second order differential equation. In general, force is a function of position and time, so using finding position as function time means the second order equation

$$F \left(x , t \right) = \frac{d^2 x}{dt^2} \left(t \right).$$

 

[Minich]

Excuse me, i suppose the question is elementary, but what is it TISE?
(Time Independent Shroedinger Equation? ))

 

[jtbell]

Yes.