THE PROBLEMYARD

Quantum Mechanics in Simple Matrix Form: chapter 5 of (2-28)

theproblemyard — Sun, 27 Feb 2011 21:31:38 +0000

: Vectors

I use “x” for vectors that are pointing farthest to the left (that is radians) and I use “y” for the other vector, or rather, a vector not pointing farthest to the left

5-1.

Find x_1 y_1 + x_2 y_2 + x_3 y_3

(a)

Given:

(b)

Given:

>
_

(c)

Given:




>
_
(d)

Given:




>

Chicken chicken chicken? …..

theproblemyard — Mon, 06 Sep 2010 05:44:20 +0000

Quantum Mechanics in Simple Matrix Form: chapter 4 of (2-28)

theproblemyard — Thu, 24 Dec 2009 07:45:03 +0000

Common matrices:

Properties:

Formula for finding inverse of matrix:

Formulas for z values in

4-1. Use the formulas for the squares and products of the Pauli matrices to calculate

Write your answers in the form

4-2. Use the formulas to find the numbers z0, z1, z2, z3 for the matrix

Formulas:

Plugging the “a” values in we get:

Check:

Test for inverse:

then there is an inverse. So we plug the z’s in:

So sorry no inverse.

4-3 Write out the 2 by 2 matrix for

multiply the matrix by itself, and thus verify that

by a second method that is different from the one already used.

They mean using matrix algebra:

The method already used was:

4-4. Use the formula to find the inverse for

and then check that your answer gives

From simplifying

we see what goes into the formula

So plugging in we get:

Check:

4-5. Find all the matrices

that commute with

We see from these two matrices that:

Therefore and

So the set of matrices that commute would be:

4-6 Use the formula to find the inverse for for

and then check that your answer gives

So we plug into the formula we saw in 4-4 for getting the inverse:

Check:

Compare this with problem 3-7.

Quantum Mechanics in Simple Matrix Form: chapter 3 of (2-28)

theproblemyard — Mon, 19 Oct 2009 07:29:06 +0000

Once again I have added letters to differentiate sub-exercises.

3-1. Find the matrices for

3-2. Compute the matrix products

3-3. Find the matrices for these products:

Again the answers should be matrices you recognize. Notice that the product to two matrices can be be zero when neither one is zero.

3-4. Show that the inverse of the matrix

A> M times M inverse>

M inverse times M>

Since we can see the products are both equal to the diagonal matrix then the M inverse we were given must be the true M inverse.

3-5. Suppose M has an inverse. Show that if A commutes with M inverse, then A commutes with M and, conversely, if A commutes with M, then A commutes with M inverse. This means there is an unambiguous matrix for A divides by M if and only if there is an unambiguous matrix AM=MA for the product of A and M.

Show that if then A commutes with M>

Show that if then A commutes with M inverse>

3-6. Let A, B, and M be matrices the same size and z a complex number. Suppose A and M commute and B and M commute. Show that A + B and M commute, that AB and M commute, and that zB and M commute.

given: and

Show that A+B and M commute>

Show that AB and M commute>

Show that zB and M commute>

(x and yi are scalers hence they give the same result whether placed before or after a matrix)

3-7. Show that the inverse of the matrix

is .

M times (1/9)M>

(1/9)M times M>

For any inverse matrix is defined as satisifying thefore (1/9)M must be the inverse of M.

Quantum Mechanics in Simple Matrix Form: chapter 2 of (2-28)

theproblemyard — Mon, 05 Oct 2009 10:07:04 +0000

Note, I added letters to help distinguish between sub-exercises. Also see my friend’s post of the same exercises.

Write all answers in the form x + iy with x and y real. (note: I didn’t write the imaginary parts like they said because I think it looks weird, instead I put the number before the i)
2-1. Work out the following sums and products:

2-2. Use the formula for to find 1/i and 1/-i and check that your answers give i(1/i)=1 and -i(1/-i)=1.

check for correctness:

Check for correctness:

2-3. ….. blah blah

2-4 Let ………..

2-5. Show that for w=3 and z=4i.

2-6. User the formula for to find and and then check that your answers give and .

For

Therefore:

Check for correctness>

2-7 Show that the following are solutions to the quadratic equation :

1st eq:

2nd eq:

I will first simplify z, and find z squared:

Therefore for z squared:

Hence we see that:

1st eq simplified:

2nd eq simplified:

1st eq squared and simplified:

2nd eq squared and simplified:

A (plugging 1st eq into )>

B (plugging 2nd eq into )>

Quantum Mechanics in Simple Matrix Form

theproblemyard — Mon, 05 Oct 2009 08:13:23 +0000

This post starts my series of problems from this book: