User:AntBap/sandbox

The Background section contains indeed several problems. Its purpose is to discuss the more familiar Euclidean vectors before moving to the more general and abstract concept of ket, which is a good approach often followed in introductory textbooks. However, a ket is a (basis-independent) vector in the sense of abstract vector spaces, not a (basis-dependent) column-vector or coordinate-vector as used in basic linear algebra. Therefore, we should not state that a Euclidean vector is an element of $$\mathbb{R}^3$$ or $$\mathbb{C}^3$$, or that $$\mathbb{C}^3$$ is a Hilbert space! This is closely related to the notation problem mentioned below, and the two should probably be addressed together. As for the question of the origin of the vectors being considered, also mentioned by the OP, it can be clarified by stating that the Euclidean vectors being discussed are free vectors, linking to the relevant article.

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Indeed, this distinction should be made clear to avoid the usual misunderstandings: a ket is a (basis-independent) vector in the sense of abstract vector spaces, not a (basis-dependent) "coordinate vector" or "column vector" as used in basic linear algebra. However, I think that some of the changes that were meanwhile introduced are incorrect, because many strict identities are now regarded as mere representations and, consequently, the symbol $$\doteq$$ is now used in several places where a plain equal symbol should be used. It seems that $$\doteq$$ has been often used in the article to indicate that a particular basis set has been chosen, when in fact it should be used only when such a choice is associated with the replacement of a basis-independent expression with a basis-dependent expression, which are two different mathematical objects that cannot be related through a strict identity. For example, the symbol $$\doteq$$ is often used too early in an equation, as in the relation
 * $$\mathbf{A} \doteq \!\, A_1 \mathbf{e}_1 + A_2 \mathbf{e}_2 + A_3 \mathbf{e}_3 = \begin{pmatrix}

A_1 \\ A_2 \\ A_3 \\ \end{pmatrix} $$ currently given in the section Background: Vector spaces, which should actually be
 * $$\mathbf{A} = \!\, A_1 \mathbf{e}_1 + A_2 \mathbf{e}_2 + A_3 \mathbf{e}_3 \doteq \begin{pmatrix}

A_1 \\ A_2 \\ A_3 \\ \end{pmatrix} $$ because the second expression is rigorously identical to the vector $$\mathbf{A}$$, while it is the last expression that corresponds to a different mathematical object which merely represents $$\mathbf{A}$$ in the particular basis set $$ \{ \mathbf{e}_1, \mathbf{e}_2, \mathbf{e}_3 \} $$. Thus, considering a second basis set $$ \{ \mathbf{e}_1', \mathbf{e}_2', \mathbf{e}_3' \} $$ gives
 * $$\mathbf{A} = \!\, A_1 \mathbf{e}_1 + A_2 \mathbf{e}_2 + A_3 \mathbf{e}_3 = \!\, A_1' \mathbf{e}_1' + A_2' \mathbf{e}_2' + A_3' \mathbf{e}_3'

$$ but

\begin{pmatrix} A_1 \\ A_2 \\ A_3 \\ \end{pmatrix} \neq \begin{pmatrix} A_1' \\ A_2' \\ A_3' \\ \end{pmatrix} $$ showing that only the "column vectors" are a basis-dependent representation. This confusion extends to the text of this section, which uses the word "representation" to refer both to the column matrix and (wrongly) to the linear combination of vectors. This too-early use of $$\doteq$$ affects all equations in this section, but not those in the section Ket notation for vectors that follows. Another problem is that the symbol $$\doteq$$ should not be used when an inner product is expressed as a sum of coordinate products, as currently done in many equations, because although the terms in the sum depend on the basis set, the expression itself does not. For example, the relation
 * $$ \langle A | B \rangle \doteq \!\, A_1^* B_1 + A_2^* B_2 + \cdots + A_N^* B_N {=}

\begin{pmatrix} A_1^* & A_2^* & \cdots & A_N^* \end{pmatrix} \begin{pmatrix} B_1 \\ B_2 \\ \vdots \\ B_N \end{pmatrix}$$ currently given in the section Bras and kets as row and column vectors should not have any $$\doteq$$ symbol, because all expressions are equal to the same complex number, namely the inner product of $$| A \rangle$$ and $$| B \rangle$$. Although the row and column matrices in the rightmost expression are themselves just a basis-dependent representation of respectively $$\langle A |$$ and $$| B \rangle$$, their matrix product is an invariant complex number strictly equal to the inner product, not a mere representation of it. In contrast, it is correct to use the symbol $$\doteq$$ for the matrix representation of outer product operators, as in the relation
 * $$ |\phi \rangle \, \langle \psi | {\doteq \!\,}

\begin{pmatrix} \phi_1 \\ \phi_2 \\ \vdots \\ \phi_N \end{pmatrix} \begin{pmatrix} \psi_1^* & \psi_2^* & \cdots & \psi_N^* \end{pmatrix} = \begin{pmatrix} \phi_1 \psi_1^* & \phi_1 \psi_2^* & \cdots & \phi_1 \psi_N^* \\ \phi_2 \psi_1^* & \phi_2 \psi_2^* & \cdots & \phi_2 \psi_N^* \\ \vdots & \vdots & \ddots & \vdots \\ \phi_N \psi_1^* & \phi_N \psi_2^* & \cdots & \phi_N \psi_N^* \end{pmatrix} $$ currently given in the section Outer products, because the elements of the matrix depend on the basis set(s) being used, making it a mere representation of the operator $$|\phi \rangle \, \langle \psi |$$. Just like a "column vector" is meaningless without indicating the basis set it refers to, a "matrix operator" is meaningless without indicating the basis set(s) it refers to. Finally, the meaning of and need for the symbol $$\doteq$$ is only explained at the end of the section Ket notation for vectors, but that should preferably be done when the symbol is first used. Overall, I think that, as it stands now, the treatment of this "identity versus representation" issue may end up confusing readers not previously aware of the distinction. This is closely related to the errors/simplifications mentioned above, and the two should probably be addressed together. If others agree with these comments, I can try to introduce the corresponding changes/corrections. AntBap (talk) 07:28, 5 May 2014 (UTC)