Quantum Mechanics Solution Manual |
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© Leon van Dommelen |
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4.4.3.1 Solution esdb2-a
Question:
The 2p pointer state of the hydrogen atom was defined as
where both and are eigenfunctions of the total energy Hamiltonian with eigenvalue and of square angular momentum with eigenvalue ; however, is an eigenfunction of angular momentum with eigenvalue , while is one with eigenvalue . Evaluate the expectation values of energy, square angular momentum, and angular momentum in the 2p state using inner products. (Of course, since 2p is already written out in terms of the eigenfunctions, there is no simplification in this case.)
Answer:
For energy you have,
By the definition of eigenfunction, the products with simplify:
Multiplying out further, while noting that on account of orthonormality of the eigenstates,
you get .
Similarly, for the square angular momentum,
or multiplying out
multiplying out further to .
For the angular momentum,
or multiplying out
multiplying out further to 0.