Every tenor man has to learn to play Body and Soul and every budding chemist has got to learn some quantum mechanics. Forget the Schrodinger equation (for now), quantum mechanics is really written in the language of linear algebra. Feynman warned us not to consider ‘how it can be like that’, but at least you can understand the ‘that’ — e.g. linear algebra. In fact, the instructor in a graduate course in abstract algebra I audited opened the linear algebra section with the remark that the only functions mathematicians really understand are the linear ones.
The definitions used (inner product, matrix multiplication, Hermitian operator) are obscure and strange. You can memorize them and mumble them as incantations when needed, or you can understand why they are the way they are and where they come from — the point of these posts. They were not handed down on tablets of stone.
In what follows, I’ll assume you have some concept of what a vector is (at least in 3 dimensional space) and what complex numbers are. It’s hard to imagine anyone studying quantum mechanics without knowing them to begin with or studying them concurrently.
If someone can tell me how to get mathematical symbols into Kubrick, the template I’m using for this, please post a comment. The guy who wrote Kubrick could not. So the notation which follows is pretty horrible.
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
X, V, etc. etc. are to be taken as vectors (I’ve got no way to put an arrow on top of them)
E1, E2, 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 (see below) can be written 3 ways V.V < V, V> and < V| V >. Since physicists use the last one, that’s what I’ll stick to.
Linear algebra concerns vector spaces, but before delving into them it’s worthwhile thinking about what linear actually means. Only two things really. A linear function f satisfies
f(a + b) = f(a) + f(b) — thing 1
# * f(a) = f(# * a) — thing 2
So what is a NONlinear function? Simple, anything with any variable raised to a power greater than 1. Such as f(x) = x^2. Not linear because
f(a+b) = (a+b)^2 = a^2 + 2 * a * b + b^2 which is not the same as
f(a) + f(b)
Thing 2 implies that if f is linear then f(0) = 0, because 0 * f(a) = 0 and f(0 * a) = f(0) = 0 * f(a)
Quantum mechanics deals in functions (called wavefunctions) and also in operators. Operators change functions into other functions (so they operate on functions the way functions operate on numbers). Differentiation and integration are good examples of operators. They are also good examples of linear operators. The derivative of the sum of two functions is the sum of the derivative of each function taken separtely. Ditto for integrals.
All operators in quantum mechanics are linear operators (this is actually one of the postulates of QM and one of the reasons understanding linear algebra is worthwhile).
The long road to matrix multiplication, eigenvectors and Hermitian matrices begins with the inner product. One can do a lot of linear algebra without them. “Linear Algebra Done Right” by Sheldon Axler doesn’t mention them for the first 99 of its 245 pages. The book is quite clear but only presents the mathematical bones of linear algebra without any physics flesh.
The definition of inner product (dot product) of a vector V with itself written < V | V>, probably came from the notion of vector length. Given the standard basis in two dimensional space E1 = (1,0) and E2 = (0,1) all vectors V can be written as x * E1 + y * E2 (x is known as the coefficient of E1). Vector length is given by the good old Pythagorean theorem as SQRT[ x^2 + y^2]. The dot product (inner product) is just x^2 + y^2 without the square root.
In 3 dimensions the distance of a point (x, y, z) from the origin is SQRT [x^2 + y^2 + z^2]. The definition of vector length (or distance) easily extends (by analogy) to n dimensions where the length of V is SQRT[x1^2 + x2^2 + . . . . + xn^2] and the dot product is x1^2 + x2^2 + . . . . + xn^2. Length is always a non-negative real number.
The definition of inner product also extends to the the dot product of two different vectors V and W where V = v1 * E1 + v2 * E2 + . … vn * En, W = w1 * E1 + . . + wn * En — e.g. V . W = v1 * w1 + v2 * w2 + . . . + vn * wn. Again always a real number, but not always positive as any of the v’s and w’s can be negative. Notice that < V | W > = < W | V > in what we’ve done so far (because we’ve assumed that all the v’s and w’s are real numbers).
However quantum mechanics deals in vectors in which the coefficients of Ei (e.g. v1, w5 etc. etc.) can be complex numbers (which are always of the form a + bi where a and b are real numbers). It’s one of the things about QM you must accept — remember Feynman — don’t ask why it has to be that way. The complex conjugate of v = a + bi is defined a – bi (written v*). Multiplying v * v* together (note the two distinct uses of *) gives a^2 + b^2, which is a nonnegative (it could be zero) real number since a and b are both real numbers.
All observables in quantum mechanics are real numbers, but the vectors representing quantum states are complex vectors (they have complex coefficients). So 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*.
This modification of the definition of dot product for complex vectors, leads to significant complications. Why? When V, W are vectors with complex coefficients < V | W > is not the same as< W | V > unlike the case when the vectors have all real coefficients.
That’s all for now. In the next post I’ll explain how quantum mechanics gets around this.
Next post — https://luysii.wordpress.com/2010/01/06/linear-algebra-survival-guide-for-quantum-mechanics-ii/