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Friday, 6 September 2013

Postulates of Quantum Mechanics: State Space

A quantum state \(\left|\psi\right>\) is an element of \emph{Hilbert space} \(\mathcal{H}\) with unit norm. A Hilbert space is a complete vector space over the complex numbers \(\mathbb{C}\) that has an inner product. This inner product allows the norms and relative angles between different states in the Hilbert space to be defined. A set of basis states can be associated to the Hilbert space \(\mathcal{H}\). If \(\mathcal{H}\) is finite dimensional with dimension \(N\), then there exists a set of \(N\) \emph{orthonormal} basis vectors \(\left|1\right>, \left|2\right>, . . . , \left|N\right>\). Orthonormaility means that the inner product between any two basis states \(\left|r\right>\) and \(\left|s\right>\) is \(0\) if \(r\not=s\), and is \(1\) if \(r=s\). Thus, every basis vector in an orthonormal basis has unit norm. In this way, an arbitrary state \(\left|\psi\right>\in\mathcal{H}\) can be expressed as a linear combination, or superposition, of basis states
 \[\left|\psi\right>= \alpha_1\left|1\right>+ \alpha_2 \left|2\right>+ . . . + \alpha_N \left|N\right>,\]
 where \(\alpha_i \in \mathbb{C}\) for all \(i\in\{1,2, . . . , N\}\) and \(\displaystyle\sum\limits_{i=1}^N |\alpha_i|^2 = 1\). This latter constraint is called the normalization constraint and implies that \(\left|\psi\right>\) is of unit norm. Such a state is said to be normalized.


Combining States


This postulate about combining states can be thought of as an extension of the first. It tells us how to construct a new state representing the joint system of multiple states using the tensor product \(\otimes\). First, the properties of the tensor products will be made precise.

Let \(\mathcal{V}\) denote the set of Hilbert spaces. Then the tensor product is a map \(\otimes: \mathcal{V}\times\mathcal{V}\rightarrow \mathcal{V}\). If \(V\) and \(W\) are \(m\) and \(n\) dimensional Hilbert spaces, respectively, then the tensor product of \(V\) with \(W\) is denoted by \(V\otimes W\) and is a Hilbert space of dimension \(mn\). An element of \(V\otimes W\) can be written in the form \(\left|v\right>\otimes \left|w\right>\), where \(\left|v\right>\in\mathcal{V}\) and \(\left|w\right>\in\mathcal{W}\). In particular, if \(\left|i\right>\in\mathcal{V}\) and \(\left|j\right>\in\mathcal{W}\) are orthonormal basis vectors of \(\mathcal{V}\) and \(\mathcal{W}\), respectively, then the tensor state \(\left|i\right>\otimes \left|j\right>\) will be an orthonormal basis vector of the space \(V\otimes W\). Hence, a general state of \(V\otimes W\) will be given as a linear combination of basis states of the form \(\left|i\right>\otimes \left|j\right>\) where \(i\in\{1,2,...,n\}\) and \(j\in\{1,2,...,m\}\). The tensor product also satisfies the following three properties.

(1) For all \(\lambda\in \mathbb{C}\), \(\left|v\right>\in\mathcal{V}\) and \(\left|w\right>\in\mathcal{W}\),
\[\lambda(\left|v\right>\otimes\left|w\right>)=(\lambda\left|v\right>)\otimes\left|w\right>=\left|v\right>\otimes(\lambda\left|w\right>)\]

(2) For all \(\left|v_1\right>, \left|v_2\right>\in\mathcal{V}\), and \(\left|w\right>\in\mathcal{W}\),

\[(\left|v_1\right>+\left|v_2\right>)\otimes\left|w\right>=\left|v_1\right>\otimes\left|w\right>+\left|v_2\right>\otimes\left|w\right>\]

(3) For all \(\left|v\right>\in\mathcal{V}\) and \(\left|w_1\right>, \left|w_2\right>\in\mathcal{W}\),

\[\left|v\right>\otimes(\left|w_1\right>\otimes\left|w_2\right>)=\left|v\right>\otimes\left|w_1\right>+\left|v\right>\otimes\left|w_2\right>\]

As a notational convenience let the following equivalently represent the tensor product of two states \(\left|v\right>\) and \(\left|w\right>\):

\[\left|v\right>\otimes\left|w\right>\equiv\left|v\right>\left|w\right>\equiv\left|vw\right>.\]

We are now finally in position to state the second axiom. Given two states
 \(\left|\psi_1\right>\in\mathcal{H}_1\) and \(\left|\psi_2\right>\in\mathcal{H}_2,\) a new state representing the joint system is given by
 \[\left|\psi\right>=\left|\psi_1\right>\otimes \left|\psi_2\right> \in\mathcal{H}.\]
 Here, \(\mathcal{H}=\mathcal{H}_1\otimes \mathcal{H}_2\) and \(dim(\mathcal{H})=dim(\mathcal{H}_1) \cdot dim(\mathcal{H}_2)\), where \(dim(\mathcal{H})\) denotes the dimension of the Hilbert space \(\mathcal{H}\). Continuing in this way, the tensor product of arbitrarily many states can be constructed.

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