Q31 P The Pauli spin matrices in quant... [FREE SOLUTION]

Chapter 3: Q31 P (page 123)

The Pauli spin matrices in quantum mechanics are $A = (\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix})$ , $B = (\begin{matrix} 0 & - i \\ i & 0 \end{matrix})$ , $C = (\begin{matrix} 1 & 0 \\ 0 & - 1 \end{matrix})$ .For the Pauli spin matrix C , find the matrices $sinkC$ , $coskC$ , $e^{kC}$ , and $e^{ikC}$ . Hint: Show that if a matrix is diagonal, say $D = (\begin{matrix} a & 0 \\ 0 & b \end{matrix})$ , then $f (D) = (\begin{matrix} f (a) & 0 \\ 0 & f (b) \end{matrix})$ .

Short Answer

Expert verified

The matrices for $sinkC$ , $coskC$ , $e^{kC}$ , and $e^{ikC}$ are role="math" localid="1658985903359" $\sin kC = (\begin{matrix} sink & 0 \\ 0 & - sink \end{matrix})$ , $cos kC = (\begin{matrix} cosk & 0 \\ 0 & cosk \end{matrix})$ , $e^{kC} = (\begin{matrix} e^{k} & 0 \\ 0 & e^{- k} \end{matrix})$ , and $e^{ikC} = (\begin{matrix} e^{ik} & 0 \\ 0 & e^{- ik} \end{matrix})$ respectively .

Step by step solution

Property of diagonal matrix:

For a diagonal matrix $M = (\begin{matrix} a & 0 \\ 0 & b \end{matrix})$ .

$\begin{array}{l} M^{n} = {(\begin{matrix} a & 0 \\ 0 & b \end{matrix})}^{n} \\ = (\begin{matrix} a^{n} & 0 \\ 0 & b^{n} \end{matrix}) \end{array}$

Taylor expansion of the function of matrix:

A Taylor series is an expansion of some function into an infinite sum of terms, where each term has a larger exponent like $x, x^{2}, x^{3}$ etc.

The Taylor expansion of any function of the matrix gives,

$f (M) = \underset{n}{鈭�} a_{n} M^{n} = (\begin{matrix} \underset{n}{鈭�} a_{n} a^{n} & 0 \\ 0 & \underset{n}{鈭�} a_{n} b^{n} \end{matrix}) = (\begin{matrix} f (a) & 0 \\ 0 & f (b) \end{matrix})$

Matrices sinkC , coskC , ekC , and eikC :

The Pauli spin matrixC is $C = (\begin{matrix} 1 & 0 \\ 0 & - 1 \end{matrix})$ .

Matrices $sinkC$ , $\cos kC$ , $e^{kC}$ , and $e^{ikC}$ are to be evaluated.

Apply Taylor expansion for matrix C .

First define $sinkC$ :

$\sin kC = \sin k (\begin{matrix} 1 & 0 \\ 0 & - 1 \end{matrix}) = (\begin{matrix} \sin k & 0 \\ 0 & - \sin k \end{matrix})$

Now, define $coskC$ :

$coskC = cosk (\begin{matrix} 1 & 0 \\ 0 & - 1 \end{matrix}) = (\begin{matrix} cosk & 0 \\ 0 & - cosk \end{matrix})$

Determine $e^{kC}$ as below.

$e^{kC} = e^{k} (\begin{matrix} 1 & 0 \\ 0 & - 1 \end{matrix}) = (\begin{matrix} e^{k} & 0 \\ 0 & - e^{k} \end{matrix})$

Calculate $e^{ikC}$ as follow.

$e^{ikC} = e^{ik} (\begin{matrix} 1 & 0 \\ 0 & - 1 \end{matrix}) = (\begin{matrix} e^{ik} & 0 \\ 0 & - e^{ik} \end{matrix})$

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Short Answer

Step by step solution

Property of diagonal matrix:

Taylor expansion of the function of matrix:

Matrices sinkC , coskC , ekC , and eikC :

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The Pauli spin matrices in quantum mechanics areA=(1001) ,B=(0-ii0) ,C=(100-1) .For the Pauli spin matrix C , find the matricessinkC ,coskC ,ekC, andeikC . Hint: Show that if a matrix is diagonal, sayD=(a00b), then f(D)=(f(a)00fb).

Short Answer

Step by step solution

Property of diagonal matrix:

Taylor expansion of the function of matrix:

Matrices sinkC , coskC , ekC , and eikC :

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Mechanics and Materials

Electricity and Magnetism

Rotational Dynamics

Translational Dynamics

Turning Points in Physics

Magnetism and Electromagnetic Induction

Study anywhere. Anytime. Across all devices.