Download Exercise Physics Engineering and more Exams Physics in PDF only on Docsity!
PHY 444. Quantum Mechanics Bra-Ket Manipulation
Dr. R. L. Herman Fall 2018
Matrix Representations - Changing Bases
1 State Vectors
The main goal is to represent states and operators in different basis. We first use brute force methods
for relating basis vectors in one representation in terms of another one. Then we will show the equivalent
transformations using matrix operations.
1.1 Inserting the Identity Operator
We begin by using the identity operator in the Sz -basis,
I = |+z〉 〈+z| + |−z〉 〈−z| , (1)
to derive matrix representations.
We first note that in the Sz -basis the basis states in the Sx and Sy -bases are given by In order to relate
|+x〉 = √^1 2 (|+z〉 + |−z〉〉) , 〈+x| = √^1 2 (〈+z| + 〈−z|〉) ,
|−x〉 = √^1 2 (|+z〉 − |−z〉〉) , 〈−x| = √^1 2 (〈+z| − 〈−z|〉) ,
|+y〉 = √^1 2 (|+z〉 + i|−z〉〉) , 〈+y| = √^1 2 (〈+z| − i〈−z|〉) ,
|−y〉 = √^1 2
(|+z〉 − i|−z〉〉) , 〈−y| = √^1 2
(〈+z| + i〈−z|〉).
|+x〉 in the Sz and Sy bases, we seek a relation between the column representations
( 〈+z|+x〉 〈−z|+x〉
z
and
〈+y|+x〉 〈−y|+x〉
x
We first consider the amplitude 〈+y|+x〉. Inserting the Sz -basis identity operator (1), we obtain
〈+y|+x〉 = 〈+y| I |+x〉
= 〈+y| (|+z〉 〈+z| + |−z〉 〈−z|) |+x〉
= 〈+y|+z〉 〈+z|+x〉 + 〈+y|−z〉 〈−z|+x〉. (2)
Similarly, we have
〈−y|+x〉 = 〈−y|+z〉 〈+z|+x〉 + 〈−y|−z〉 〈−z|+x〉. (3)
Recalling that the goal is to relate |+x〉 in the Sy and Sz bases, we compute
〈+z|+x〉 =
, 〈+y|+z〉 =
〈+z|−x〉 =
, 〈+y|−z〉 = −
i √ 2
〈−z|+x〉 =
, 〈−y|+z〉 =
〈−z|−x〉 = −
, 〈−y|−z〉 =
i √ 2
Therefore, we have
〈+y|+x〉 = 〈+y|+z〉 〈+z|+x〉 + 〈+y|−z〉 〈−z|+x〉
i √ 2
1 − i
2
〈−y|+x〉 = 〈−y|+z〉 〈+z|+x〉 + 〈−y|−z〉 〈−z|+x〉
i √ 2
1 + i
2
So, the matrix representation of |+x〉 in the Sy -basis is given by
|+x〉 −→ Sy
〈+y|+x〉 〈−y|+x〉
1 − i 1 + i
1.2 A Matrix Approach
A more elegant approach is to write the equations
〈+y|+x〉 = 〈+y|+z〉 〈+z|+x〉 + 〈+y|−z〉 〈−z|+x〉.
〈−y|+x〉 = 〈−y|+z〉 〈+z|+x〉 + 〈−y|−z〉 〈−z|+x〉.
in matrix form. (^) ( 〈+y|+x〉 〈−y|+x〉
〈+y|+z〉 〈+y|−z〉 〈−y|+z〉 〈−y|−z〉
〈+z|+x〉 〈−z|+x〉
Evaluating the matrices on the right side of the equation, we obtain
( 〈+y|+x〉 〈−y|+x〉
1 −i 1 i
1 − i 1 + i
Then, we write
S =
Therefore,
S
†
Then,
|+y〉 −→ Sx
i
1 + i 1 − i
2 Operators in New Bases
We can use rotations to represent operators, A,ˆ in the Sy basis. In general, we compute matrix elements of
the matrix representation of the operator, (^) y 〈φ| Aˆ|ψ〉y , by using the identity
I = Rˆ Rˆ
† = Rˆ
R
and the change of representations of states. Namely, we have
|ψ〉y = Rˆ † |ψ〉z , |ψ〉z = Rˆ |ψ〉y ,
to obtain a general expression for matrix elements of Aˆ :
y 〈φ|^ Aˆ|ψ〉 y =^ y^ 〈φ|^ Rˆ Rˆ†^ Aˆ Rˆ†^ Rˆ|ψ〉 y
= (^) z 〈φ|R
A Rˆ|ψ〉z (8)
This means that
Aˆ −→ Sy
S
† AS where Aˆ −→ Sz
A. (9)
In other words, we compute S † AS in the Sz -basis to obtain the representation of Aˆ in the Sy -basis.
Example Represent Jˆz in the Sy -basis.
We know that Jˆz is diagonal in the Sz -basis. So,
Jˆ
z −→ Sz
Jz =
No, we compute the matrix representation of Jˆz in the Sy -basis using the similarity transformation,
S † Jz S,
J^ ˆ
z −→ Sy
S
† Jz S =
1 −i 1 i
i −i
1 −i 1 i
−i i
