Quantum Mechanics Solution Manual |
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© Leon van Dommelen |
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4.1.5.1 Solution harme-a
Question:
Just to check that this book is not lying, (you cannot be too careful), write down the analytical expression for and using table 4.1. Next write down and . Verify that the latter two are the functions and in a coordinate system that is rotated 45 degrees counter-clockwise around the -axis compared to the original coordinate system.
Answer:
Take the rotated coordinates to be and as shown:
A vector displacement of magnitude in the -direction has a component along the -axis of magnitude , equivalent to . Similarly, a vector displacement of magnitude in the -direction has a component along the -axis of magnitude , equivalent to . So in general, for any point ,
Similarly you get
Turning now to the eigenfunctions, taking the generic expression
and substituting 1, 0, you get
Now substitute for and from table 4.1:
where the constant is as given in table 4.1. You can multiply out the exponentials:
The same way, you get
So, the combination is
Now is according to the Pythagorean theorem the square distance from the origin, which is the same as . And since , the sum in the combination eigenfunction above is . So the combination eigenfunction is
which is exactly the same as above, except in terms of and . So it is in the rotated frame.
The other combination goes the same way.