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How to Derive the Schrödinger Equation

In quantum physics, the Schrödinger technique, which involves wave mechanics, uses wave functions, mostly in the position basis, to reduce questions in quantum physics to a differential equation.

Werner Heisenberg developed the matrix-oriented view of quantum physics, sometimes called matrix mechanics. The matrix representation is fine for many problems, but sometimes you have to go past it, as you’re about to see.

One of the central problems of quantum mechanics is to calculate the energy levels of a system. The energy operator is called the Hamiltonian, H, and finding the energy levels of a system breaks down to finding the eigenvalues of the problem:

image0.png

Here, E is an eigenvalue of the H operator.

Here’s the same equation in matrix terms:

image1.png

The allowable energy levels of the physical system are the eigenvalues E, which satisfy this equation. These can be found by solving the characteristic polynomial, which derives from setting the determinant of the above matrix to zero, like so

image2.png

That’s fine if you have a discrete basis of eigenvectors — if the number of energy states is finite. But what if the number of energy states is infinite? In that case, you can no longer use a discrete basis for your operators and bras and kets — you use a continuous basis.

Representing quantum mechanics in a continuous basis is an invention of the physicist Erwin Schrödinger. In the continuous basis, summations become integrals. For example, take the following relation, where I is the identity matrix:

image3.png

It becomes the following:

image4.png

And every ket

image5.png

can be expanded in a basis of other kets,

image6.png

like this:

image7.png

Take a look at the position operator, R, in a continuous basis. Applying this operator gives you r, the position vector:

image8.png

In this equation, applying the position operator to a state vector returns the locations, r, that a particle may be found at. You can expand any ket in the position basis like this:

image9.png

And this becomes

image10.png

Here’s a very important thing to understand:

image11.png

is the wave function for the state vector

image12.png

— it’s the ket’s representation in the position basis.

Or in common terms, it’s just a function where the quantity

image13.png

represents the probability that the particle will be found in the region d3r centered at r.

The wave function is the foundation of what’s called wave mechanics, as opposed to matrix mechanics. What’s important to realize is that when you talk about representing physical systems in wave mechanics, you don’t use the basis-less bras and kets of matrix mechanics; rather, you usually use the wave function — that is, bras and kets in the position basis.

Therefore, you go from talking about

image14.png

This wave function is just a ket in the position basis. So in wave mechanics,

image15.png

becomes the following:

image16.png

You can write this as the following:

image17.png

But what is

image18.png

It’s equal to

image19.png

The Hamiltonian operator, H, is the total energy of the system, kinetic (p2/2m) plus potential (V(r)) so you get the following equation:

image20.png

But the momentum operator is

image21.png

Therefore, substituting the momentum operator for p gives you this:

image22.png

Using the Laplacian operator, you get this equation:

image23.png

You can rewrite this equation as the following (called the Schrödinger equation):

image24.png

So in the wave mechanics view of quantum physics, you’re now working with a differential equation instead of multiple matrices of elements. This all came from working in the position basis,

image25.png

When you solve the Schrödinger equation for

image26.png

you can find the allowed energy states for a physical system, as well as the probability that the system will be in a certain position state.

Note that, besides wave functions in the position basis, you can also give a wave function in the momentum basis,

image27.png

or in any number of other bases.

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