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How to Find the Second-Order Corrections to Energy Levels and Wave Functions

How to Find the First-Order Corrections to Energy Levels and Wave Functions

In quantum physics, in order to find the first-order corrections to energy levels and wave functions of a perturbed system, En, you need to calculate E(1)n, as well as

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So how do you do that? You start with three perturbed equations:

image1.png

You then combine these three equations to get this jumbo equation:

image2.png

You can handle the jumbo equation by setting the coefficients of lambda on either side of the equal sign to each other. After matching the coefficients of lambda and simplifying, you can find the first-order correction to the energy, E(1)n, by multiplying

image3.png

Then the first term can be neglected and you can use simplification to write the first-order energy perturbation as:

image4.png

Swell, that's the expression you use for the first-order correction, E(1)n.

Now look into finding the first-order correction to the wave function,

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You can multiply the wave-function equation by this next expression, which is equal to 1:

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So you have

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Note that the m = n term is zero because

image8.png

So what is

image9.png

You can find out by multiplying the first-order correction,

image10.png

And substituting that into

image11.png

gives you

image12.png

Okay, that's your term for the first-order correction to the wave function,

image13.png

The wave function looks like this, made up of zeroth-, first-, and second-order corrections:

image14.png

Ignoring the second-order correction and substituting

image15.png

in for the first-order correction gives you this for the wave function of the perturbed system, to the first order:

image16.png
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