Fund Quantum Mechanics Lect & HW Solutions 67

Fund Quantum Mechanics Lect & HW Solutions 67 - a L...

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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 coeFcients 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 a E A = 1 2 E 2 + 1 2 E 2 = E 2 = 3 . 4 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, a L 2 A = 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
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Unformatted text preview: a L z A = 1 2 h 1 2 h = 0 Measurements in which the z-angular momentum is found to be h average out against those where it is found to be h , 4.3.2.2 Solution esdb-b Question: Continuing the previous question, what are the standard deviations in energy, square angular momentum, and z-angular momentum? Answer: Since the expectation value in energy is E 2 , as are the eigenvalues of each state, the standard deviation is zero. E = r 1 2 ( E 2 E 2 ) 2 + 1 2 ( E 2 E 2 ) 2 = 0 . This is expected since every measurement produces E 2 ; there is no deviation from that value. Similarly the standard deviation in L 2 is zero: L 2 = r 1 2 (2 h 2 2 h 2 ) 2 + 1 2 (2 h 2 2 h 2 ) 2 = 0 ....
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This note was uploaded on 01/06/2012 for the course PHY 3604 taught by Professor Dr.danielarenas during the Fall '11 term at UNF.

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