
Physical Chemistry
2nd Edition
ISBN: 9781133958437
Author: Ball, David W. (david Warren), BAER, Tomas
Publisher: Wadsworth Cengage Learning,
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Transcribed Image Text:A two-level system is in a quantum state = α₁₁ + a22, which can be
represented by the vector a = {a1, a2}. We are looking for the conditions
under which is eigenstate of the operator c., defined below.
1) In the expression c.ỗ, c is a vector with components {Cr, Cy, Cz} ( C; are
real numbers) and ỗ is a vector with components {σx, σy,σz} (σ¿ are 2 × 2
matrices). This means that the operator c. can also be represented by a
2 × 2 matrix. Write the matrix of the operator c. knowing that
0
0x =
= (₁ }) = ( 5 ) .
Oy
σz=
0
(6-99)
1 0
1
0
2) In matrix form, the eigenvalue equation is AX AX, where A is the
matrix of the operator of interest, X the column matrix representing the
eigenvector and the corresponding eigenvalue. Write the eigenvalue equa-
tion that needs to be verified for the quantum state to be an eigenvector
of the operator c..
=
3) Note that AX = XX ⇒ (A - I)X 0, where I is the identity matrix.
(AAI)X=0 is true if and only if det(A - I) = 0. Solve this equation for
the operator c. and write the eigenvalues as a function of Cx, Cy and Cz.
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