Linear Algebra survival guide for Quantum Mechanics – II

Before pushing on to the complexities of the dot product of two complex vectors, it’s worthwhile thinking about why the dot product isn’t a product as we’ve come to know products.  Consider E1 = (1, 0 ) and E2 = (0, 1).  Their dot product is 1 * 0 + 0 * 1 or zero.  Not your father’s product.  You’re not in Kansas any more.  Abstract algebraists, love such things and call them zero divisors, because neither of them is zero themselves yet when ‘multiplied’ together they produce zero.

This is not just mathematical trivia, as any two vectors we can dot together and get zero are called orthogonal. Such vectors are particularly important  for quantum mechanics, because (to get ahead a bit) all the eigenvectors we can interrogate by experiment to get any sort of measurement (energy, angular momentum etc. etc.)  are orthogonal to each other.   The dot product of V = 3 * E1 + 4 * E2 with itself is 25.    We can can make < V | V > = 1 by multiplying V by 1/SQRT(25)  — check it out.  Such a vector is said to be normalized.  Any vector you meet in quantum mechanics can and should be normalized, and usually is, except on your homework, where you forgot to do it and got the wrong answer. Vectors which are both orthogonal to each other and normalized are called (unsurprisingly) orthonormal.

I’d love to be able to put subscripts on the variables, but at this point I can’t, so here are the naming conventions once again.

x^2 means x times x (or x squared)
x1 means x with subscript 1 (when x is a small letter)
x57 (note two integers follow the x not one) means a matrix element with the first number for the Row and the second for the Column — mnemonic Roman Catholic
XV, etc. etc. are to be taken as vectors (I’ve got no way to put an arrow on top of them)
E1E2, are the standard basis vectors — E1 = (1, 0 , 0 . . ), E2 = (0, 1, 0, .. ), En = (0, 0, … 1), Ei stands for any of them
# stands for any number (which can be real or complex)
i (in italics) always stands for the SQRT[-1]
* has two meanings. When separated by spaces such as x * x it means multiply e.g. x^2
When next to a vector V* or a letter x* it means the complex conjugate of the vector or the number (see later)

The dot product of a vector V can be written 3 ways  V.VV,V> and < VV >.  Since physicists use the last one, that’s what I’ll stick to (mostly).

Recall that to get a real number from the dot product of a complex vector with itself, one must multiply the vector V by its complex conjugate V*.  Here what the complex conjugate is again.  Given a complex number z = a + ib, its complex conjugate (written z*) is a – ib.

z * z* (note the different uses of *) =   a^2 + b^2, which is a real nonNegative number because a and b are both real.  Note that conjugating a complex number twice doesn’t change it –e.g.  z** = z.

This modification of the definition of dot product for complex vectors, leads to significant complications. Why? When V, W are vectors with complex coefficients < VW > is NOT the same as < WV > unlike the case where the vectors have all real coefficients.  Here’s why.  No matter how many components a complex vector has, the dot product is only a sum of the products of just two complex numbers with each other (see the previous post).  The product of two complex numbers is just another one, as is the sum of any (finite) number of complex numbers.  This means that multiplying a mere two complex numbers together will be enough to see the problem. To avoid confusion with V and W which are vectors, I’ll call the complex numbers p and q. Remember that p1, p2, q1 and q2 are all real numbers and i is just, well i (the number which when multiplied by itself gives – 1).

p = p1 + p2i,       q = q1 + q2i

p* = p1 – p2i,      q* = q1 – q2i

p times q* = (p1 + p2i) * (q1 – q2i) = (p1 * q1 + p2 * q2) + i (p2 * q1 – p1 * q2)

p* times q = (p1 – p2i) * (q1 + q2i) =  (p1 * q1 + p2 * q2) + i(p1 * q2 – p2 * q1)

Note that the terms which multiply i are NOT the same (but they are the negative of each other).   So what does < V | W > mean?  Recall that

V = v1 * E1 + v2 * E2 + . … vn * En

W = w1 * E1 + . . + wn * En

VW > = v1 * w1 + v2 * w2 + . . . + vn * wn. ; here the * means multiplication not complex conjugation.

Remember that v1, w1, v2, etc. are now complex numbers, and you’ve just seen that v1* times w1 is NOT the same as v1 times w1*.  Clearly a convention is called for. Malheureusement, physicists use one convention and mathematicians use the other.   Since this is about quantum mechanics, here’s what physicists mean by < V | W >.  They mean the dot product of V* (whose coefficients are the complex conjugates of v1, . . v2) with W.  More explicitly they mean V* . W, but when written in physics notation < V | W >, the * isn’t mentioned (but never forget that it’s there).

Now  v1 * w1 + v2 * w2 + . . . + vn * wn is just another complex number — say z =  x + iy.  To form its complex conjugate we just negate the iy term to get z* = x – iy

Look at

p times q* = (p1 + p2i) *  (q1 – q2i) = (p1 * q1 + p2 * q2) + i (p2 * q1 – p1 * q2)

p* times q = (p1 – p2i) * (q1 + q2i) =  (p1 * q1 + p2 * q2) + i(p1 * q2 – p2 * q1)

once again.  Notice that p times q* is just the complex conjugate of p* times q

So if < V | W > = v1* * w1 + v2* * w2 + . . . + vn* * wn = x + iy ;  here * means 2 different things, complex conjugation when next to vi and multiplication when between vi and wi (sorry for the horrible notation, hopefully  someone knows how to get subscripts into all this).

By the physics convention < W | V > is w1* * v1 + w2* * v2 + . . . + wn* * vn.  Since p times q* is just the complex conjugate of p* times q,  w1* * v1 is the complex conjugate of w1 * v1*.  This means  w1* * v1 + w2* * v2 + . . . + wn* * vn = x – iy.

In shorthand < V | W > = < W | V >*, something you may have seen and puzzled over.  It’s all a result of wanting the dot product of a complex vector to be a real number.  Not handed down on tablets of stone, but the response to a problem.

Next up, vector spaces, linear transformations on them (operators) and their matrix representation.  I hope to pump subsequent posts  out one after the other, but I’m having some minor surgery on the 6th, so there may be a lag.

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Comments

  • correction  On February 16, 2011 at 4:58 am

    In the line,
    p* times q = (p1 – p2i) * (q1 + q2i) = (p1 * q1 + p2 * q2) + i(q1 * q2 – p2 * q1),

    the last part should be i(p1 * q2 – p2 * q1)?

  • luysii  On February 16, 2011 at 8:50 pm

    You’re absolutely right. I’ve changed it — thanks.

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