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Quantum Engi Q and A_Part_17

# Quantum Engi Q and A_Part_17 - 4.3 EXPECTATION VALUE AND...

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4.3. EXPECTATION VALUE AND STANDARD DEVIATION 49 What are the expectation values of energy, square angular momentum, and z -angular momen- tum for this state? Answer: Note that the square coefficients of the eigenfunctions ψ 211 and ψ 21 1 are each 1 2 , so each has a probability 1 2 in the 2p x state. Eigenfunction ψ 211 has an energy eigenvalue E 2 , and so does ψ 21 1 , so the expectation value of energy in the 2p x state is ( E ) = 1 2 E 2 + 1 2 E 2 = E 2 = 4 . 3 eV. This is as expected since the only value that can be measured in this state is E 2 . Similarly, eigenfunction ψ 211 has a square angular momentum eigenvalue 2¯ h 2 , and so does ψ 21 1 , so the expectation value of square angular momentum in the 2p x state is that value, ( L 2 ) = 1 2 h 2 + 1 2 h 2 = 2¯ h 2 . Eigenfunction ψ 211 has a z -angular momentum eigenvalue ¯ h , and ψ 21 1 has ¯ h , so the expec- tation value of z -angular momentum in the 2p x state is ( L z ) = 1 2 ¯ h 1 2 ¯ h = 0 Measurements in which the z -angular momentum is found to be ¯ h average out against those

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