Understanding the Laplacian Watch

Joseph Mckeown
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I've recently been trying to get my head around the Schrödinger equation, considering a single particle.

I know that the Hamiltonian can be given by
H = T + V

Where V is the potential energy, V(r, t)

And T is the total kinetic energy, given in terms of momentum...

T = (p.p)/2m

Which becomes, on multiplying out...

T = (-ħ/2m) × ∇^2

Where ∇^2 can be given in three dimensions as the sum of the three partial derivatives with respect to each of the directions, x, y and z.

What I don't understand is the following: what are the partial derivatives derived from? What function am I to differentiate to arrive at them?

In simple terms, what IS the Laplacian?

Thank you!
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atsruser
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(Original post by Joseph Mckeown)
What I don't understand is the following: what are the partial derivatives derived from? What function am I to differentiate to arrive at them?

In simple terms, what IS the Laplacian?
You seem not to be aware that the T+V formulation is a differential *operator*, to be applied to a function, namely the wave function, usually written as \psi(x,y,z,t). The wave function contains all of the information that describes the quantum system of interest (like an atom or molecule).

When you consider a real 3D system (such as an atom), you need to deal with all 3 spatial dimensions i.e. you have x,y,z coords in your wave function and T+V operator, but it's easier to start with toy 1D systems, to see how the Schroedinger equation works. Then the Laplacian reduces to \partial^2/\partial x^2 and it operates on a 1D wave function \psi(x,t).

You can google "1D particle in a box" or "1D harmonic oscillator" for examples.

When you are happy with solving the S. eqn in 1D, then it makes sense to start looking at 3D systems (which are mathematically much trickier).
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Joseph Mckeown
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(Original post by atsruser)
You seem not to be aware that the T+V formulation is a differential *operator*, to be applied to a function, namely the wave function, usually written as \psi(x,y,z,t). The wave function contains all of the information that describes the quantum system of interest (like an atom or molecule).

When you consider a real 3D system (such as an atom), you need to deal with all 3 spatial dimensions i.e. you have x,y,z coords in your wave function and T+V operator, but it's easier to start with toy 1D systems, to see how the Schroedinger equation works. Then the Laplacian reduces to \partial^2/\partial x^2 and it operates on a 1D wave function \psi(x,t).

You can google "1D particle in a box" or "1D harmonic oscillator" for examples.

When you are happy with solving the S. eqn in 1D, then it makes sense to start looking at 3D systems (which are mathematically much trickier).
That's fantastic, thank you! I was aware that the Hamiltonian is an operator, I just hadn't joined the dots to that meaning that the partial derivatives would correspond to the wavefunction in question. All part of the learning curve, I guess!

Thanks again for your help
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atsruser
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(Original post by Joseph Mckeown)
That's fantastic, thank you! I was aware that the Hamiltonian is an operator, I just hadn't joined the dots to that meaning that the partial derivatives would correspond to the wavefunction in question. All part of the learning curve, I guess!

Thanks again for your help
Right. If we write it out in full, we get a differential equation with this in it:

\frac{\partial^2 \psi}{\partial^2 x}+\frac{\partial^2 \psi}{\partial^2 y} + \frac{\partial^2 \psi}{\partial^2 z}

which we normally write as \nabla^2 \psi when we don't want to indicate which kind of coord system we are using.

Often it is much easier to solve the S. eqn when we write it in spherical polar coords, since this corresponds to a physical symmetry of the system e.g. the potential term in the hydrogen atom is spherically symmetric so depends only on r. When you express the Laplacian in spherical coords, you need to do the same for the wave function, of course, so you would then write it as \psi(r,\theta,\phi,t).

You therefore need to know how to express the Laplacian in various different coord systems. Google is your friend for details of course.
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