Solutions4.pdf
4
Quantum Mechanics I Solutions 4. HS 2012 Prof. Ch. Anastasiou Exercise 1. Ener gy eigenstates Let us consider an hypothetical two-state system described by the following Hamiltonian H = a (|1 1| − | 2 2| + |1 2| + |2 1|) , where a is a numbe r with the dimension of energy. Find the energy eigen v alues and their corresponding energy eigenkets, in the | 1 & | 2 basis. Solution. The best wa y to proceed is to wr ite H in matrix representation in the |1 & |2 basis H = 1|H |1 1|H |2 2|H |1 2|H |2 = a 1 1 1 −1 . Then the eigenvalues are the roots λ i of the characteristic polynomial det( H − λ1) and the corresponding eigen- vectors are found by solving H vi = λivi . We get λ ± = ± √ 2a , v ± = 1 ± √ 2 1 = 1 ± √ 2 |1 + |2 . Exerci se 2. Spin eigenstates in arbitr ary dir ecti on Construct | S · ˆ n, + such that S · ˆ n |S · ˆ n, + = 2 | S · ˆ n, + , (1) where S = (S x ,S y ,S z ) and ˆ n is characterized by α, the polar angle, and β the azimuthal one. Express your answer as a linear combination of | + and |− . Hint. The answer is: cos( β 2 ) |+ + sin( β 2 )e iα |− . Solution. The solutio n to this exercise is atta ched on pa ge 3. Exerci se 3. Comp atible Op era tors Consider a three-dimensional ket space, spanned by the set of orthonormal kets | 1, | 2 and | 3. The operators A a nd B act on them as follows ( a, b ∈ R): A |1 = a |1 , B |1 = b |1 , A |2 = −a |2 , B |2 = ib |3 , (2) A |3 = −a |3 , B |3 = −ib |2 . (a) Obv iously A exhibits a degenerate spectrum. Does B also exhibit a degen erate spectrum? 1
-
Upload
shweta-sridhar -
Category
Documents
-
view
213 -
download
0
Transcript of Solutions4.pdf
7/27/2019 Solutions4.pdf
http://slidepdf.com/reader/full/solutions4pdf 1/4
7/27/2019 Solutions4.pdf
http://slidepdf.com/reader/full/solutions4pdf 2/4
7/27/2019 Solutions4.pdf
http://slidepdf.com/reader/full/solutions4pdf 3/4
7/27/2019 Solutions4.pdf
http://slidepdf.com/reader/full/solutions4pdf 4/4
