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Part of The Principle Of Superposition

First we will lay out some general properties of these vectors, which can be deduced from the associated axioms: it is also desirable to have a name for the class of vectors which are connected to the states of dynamical systems found within quantum mechanics, whether they are in a finite or infinite space.

The term for these vectors will be kets, and they will be denoted by the symbol $|\rangle$ , and they can be specified with labels inserted between the two symbols: $|A\rangle$ .

Ket vectors can be multiplied by complex numbers, and added together to give new ket vectors:

$c_1|A\rangle + c_2 |B\rangle = |R\rangle$ , where $c_1$ and $c_2$ are complex numbers.

In addition to this, we can perform more general linear processes, such as summing an infinite sequence of them, and given a ket vector $|x\rangle$ , dependent upon some parameter x, and the values the parameter can take, then we can integrate it with respect to x, to get another vector: $\int |x\rangle dx = |Q\rangle$ .

A ket vector which can be described linearly in terms of other vectors is said to be dependent upon said vectors, and are referred to as independent if this cannot be done.

"...each state of a dynamical system at a particular time corresponds to a ket vectors, the correspondence being such that if a state results from the superposition of certain other states, its corresponding ket vector is expressible linearly in terms of the corresponding ket vectors of the other states, and conversely."

Thus, the state $R$ is a superposition of the states $A$ and $B$ , when the vectors are connected by equation 1.

From the preceding assumptions follow certain properties of the process of superposing states (or vectors): when two more states are superposed, the order in which this is done is irrelevant, so the process is symmetrical between the superposed states.

Again we see from equation 1 (save from the case where one of the coefficients $c_1, c_2$ are zero), if $R$ can be formed from superposing states $A, B$ , then, $A$ can logically be formed from a superposition of $B,R$ , and $B$ , from $A,R$ .

equation 1 $=$

$c_1|R\rangle + c_2|B\rangle = |A\rangle =$

$c_1|A\rangle + c_2|R\rangle = |B\rangle$

A state which is the result of superposing two states, is dependent upon those states, if the ket vector corresponding is dependent upon the vectors of the set of states, and can be said to be independent if no one of them is dependent upon the other.

Referenced in

Quantum Information Theory

A subspace of $V$ is any subset of $V$ , which is closed under addition and multiplication by complex numbers.

The Principle Of Superposition