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I want to take a simple plot and change the x-axis so it appears to be measured in terms of certain symbols that are used in the physics literature. See the image below for an example.

I will rephrase this as follows: I have some set of numerical coordinates which trace out the curves in my plot. For equal spacings on the x-axis (say once every 100 points), I want to label the line $x=100,n$ with a symbol.

I don't know how to do this. The documentation provided by Mathematica didn't help me. This doesn't seem like something done by just changing the ticks settings.

This can be done using either Ticks if you're using axes or FrameTicks if you're using a frame on your plot. I made up a plot since I'm not sure the exact data matters.

Most of the code below is flair to make the graph look a bit nicer. The important bit is FrameTicks. I've told MMA to use its best judgement for 3 of the 4 sides of the graph. For plots, the order is usually left, right, bottom, top, though for certain things you can get away with only 2 arguments x-argument, y-argument.

For each side of the frame, FrameTicks is expecting a list of ticks and the label to put on those ticks, so in place of bottom from the above list, I would put something like x-value1, "x-label1", x-value2, "x-label2", .... It is also possible to specify the lengths of the ticks in this way: x-value1, "x-label1", insidelength1, outsidelength1, x-value2, "x-label2", insidelength2, outsidelength2, ....

Plot[
 Piecewise[
 -(x - 5)^2 + 50, 0 <= x <= 10,
 -(x - 10)^2 + 25, 10 < x < 15],
 x, 0, 15,
 Axes -> False,
 Frame -> True, False, True, False,
 FrameLabel -> "E (eV)", None, "k-space", None,
 FrameStyle -> Directive[16, Black],
 FrameTicks -> 
 Automatic, Automatic, 0, "K", 5, "Γ", 10, "M", 15, "K", Automatic,
 ImageSize -> 500,
 Epilog -> 
 Dashing[0.001, 0.01],
 Line[0, -1, 0, 25],
 Line[5, -1, 5, 50],
 Line[10, -1, 10, 25],
 Line[15, -1, 15, 0]
 
 ]