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Transcript
PC3130/3201 QUANTUM MECHANICS TUTORIAL 3 1. Consider a physical system with only two basic states 1 1 > = 0 and 0 2 > = . 1 The most general state of this system can be written as > = c1 1 > + c2 2 > where c1 and c2 are in general complex numbers. If the Hamiltonian is given by Eo A H= A Eo where E0 and A are real constants, obtain the eigenvalues and eignstates of the Hamiltonian by solving the Schrödinger time-independent equation H > = E > . 2. Using [ i , j] = i ijk k , show that [ , 2] = 0. ~ 3. ~ If a and b are any two vectors (or two vector operators that commute with ~ ~ each and also with the orbital angular momentum ), show that ~ [ a · , b · ] = i ( a b )· ~ 4. ~ ~ ~ ~ ~ ~ Show that if a physical system is in an eigenstate of 1 , then the expectation values of 2 and 3 vanish. Here i is the ith component of the orbital angular momentum operator . ~ 5. The Hamiltonian H of a particle moving in a central potential field is given by H p 2 ~ 2m 2 V ( r ), r x ~ Show that the orbital angular momentum commutes with the above ~ Hamiltonian, that is, i , H 0 , 6. i 1, 2, 3. An electron is in the spin state 3i A . 4 (i) (ii) (iii) 7. Determine the normalisation constant A . If you measure S3 , what is the probability of getting / 2 ? Find the expectation values of S1, S2, and S3 . (i) Find the eigenvalues and eigenstates of S2 . (ii) If you measured S2 on a particle in the general state a b what values might you get, and what is the probability of each? (iii)If you measured probabilities? S 22 , what values might you get and with what
