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Vector Math for Quantum Mechanics

This is not to be mistaken for any kind of formal math, such as may be found in linear algebra. This is a quick and dirty summary with a narrow focus.

Quantum mechanics most often uses a vector space over the complex numbers. Classical physics tends to use vectors over the real numbers. Because the real numbers are a subset of the complex numbers classic vectors are just a special case of vectors over the complex numbers. The problem is, in some cases, the properties of classical vectors do not hold with more general vectors.

Quick Take Away

Vectors represent states in a system.

Bra and Ket notation (Dirac Bra and Ket)

{$ \large \left| a \right\rangle $} is the ket a which represent a column vector.

{$$ \large |a \rangle = \begin{pmatrix} a_1 \\ a_2 \\ \dots \\ a_n \end{pmatrix} $$}

The elements of the column vector are complex numbers because this is a vector space over the complex field. The number of elements in a column is the dimensionally of the vector. Illustrated above is an n-dimension vector.

Row vectors are represented by a bra, which is the reverse of the ket:

{$$ \large \langle b| = \begin{pmatrix} b_1 \; b_2 \; \dots \; b_n \end{pmatrix} $$}

(There are some texts and lectures which have bras as column vectors and kets as row vectors, but presented here is what seems to be prevailing usage at this time.)

Whatever is between the bar and the angle bracket in a bra or a ket is just a label. Labels can be completely arbitrary, although in some contexts there are widely used meanings for particular label. The point is, the labels are not used for computation. Often lectures and author use labels loosely, in a "you know what I mean" way.

For example: {$ | 1 \rangle $} may just be the name of a state which is opposed to an alternate state {$ | 0 \rangle $}, or it may represent the vector which has 1 as the first element and 0 for other elements:

{$$ \large |1 \rangle = \begin{pmatrix} 1 \\ 0 \\ \dots \\ 0 \end{pmatrix} $$}

(which is also often labeled {$ | e_1 \rangle $}), or it might simply be the first of a number of examples. What it does not mean is the numerical value of the integer 1.

The conjugate of a vector is simply a vector whose elements are replaced by their complex conjugates:

{$$ \large |a \rangle^\star = \begin{pmatrix} a_1^\star \\ a_2^\star \\ \dots \\ a_n^{\star} \end{pmatrix} $$}

The transpose of the conjugate vector (or the conjugate of the transpose) is the bra of the vector.

{$$ \large \langle a| = |a \rangle^\dagger \\ \large \langle a|^\dagger = |a \rangle $$}

In other words, for every vector {$ \large \left| a \right\rangle $} there is a complex conjugate {$ \large \left\langle a \right| $} which is known as bra. Bras represent row vectors which consist of the complex conjugates of the elements of {$ \large \left| a \right\rangle $}.

{$$ \large |a \rangle = \begin{pmatrix} a_1 \\ a_2 \\ \dots \\ a_n \end{pmatrix} \iff \\ \left| a \right\rangle^\dagger = \left( \matrix{a_1^{\star} \; a_2^{\star} \; \dots \; a^{\star}_n} \right) = \left\langle a \right| $$}

Math properties of vector:

  1. A vector can be multiplied by a scalar to yield another vector. The scalars may be complex numbers because this is a vector space over the complex field. Since the complex numbers are a superset of the real numbers, it should be obvious that vector spaces over the real number are a subset of vector spaces over the complex numbers.
  2. Two vectors may be added to yield a resulting vector by summing the corresponding elements. Naturally, this is only defined when the vectors are of the same dimension, meaning when they have the same number of elements.
  3. Scalar multiplication is distributive over vector addition.

Some properties and definitions which are intuitively obvious.

  1. Vectors are equal if they have the same dimensionality and their corresponding elements are equal.
  2. The scalar 1 is the identity element for scalar multiplication.
  3. The identity vector for vector addition is the zero vector, which can be thought of as a vector with all zero elements.

Complex Numbers Review

{$$ \large e^{i\theta} = \cos\theta + i \sin\theta $$}




Category: Physics

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This is a student's notebook. I am not responsible if you copy it for homework, and it turns out to be wrong.

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December 24, 2018

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