1.4: The Geometry of Hilbert Space
- Page ID
- 92689
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We saw above that the quantum state of the \(k\)-state system is described by a sequence of \(k\) complex numbers \(\alpha_{0}, \ldots, \alpha_{k-1} \in \mathbb{C}\), normalized so that \(\sum_{j}\left|\alpha_{j}\right|^{2}=1\). So it is natural to write the state of the system as a \(k\) dimen-
sional vector:
\[
|\psi\rangle=\left(\begin{array}{c}
\alpha_{0} \\
\alpha_{1} \\
\vdots \\
\alpha_{k-1}
\end{array}\right)
\]
The normalization on the complex amplitudes means that the state of the system is a unit vector in a \(k\) dimensional complex vector space - called a Hilbert space.
But hold on! Earlier we wrote the quantum state in a very different (and simpler) way as: \(\alpha_{0}|0\rangle+\alpha_{1}|1\rangle+\cdots+\alpha_{k-1}|k-1\rangle\). Actually this notation, called Dirac's ket notation, is just another way of writing a vector. Thus
\[
|0\rangle=\left(\begin{array}{c}
1 \\
0 \\
\vdots \\
0
\end{array}\right), \quad|k-1\rangle=\left(\begin{array}{c}
0 \\
0 \\
\vdots \\
1
\end{array}\right)
\]
So we have an underlying geometry to the possible states of a quantum system: the \(k\) distinguishable (classical) states \(|0\rangle, \ldots,|k-1\rangle\) are represented by mutually orthogonal unit vectors in a \(k\)-dimensional complex vector space. i.e. they form an orthonormal basis for that space (called the standard basis). Moreover, given any two states, \(\alpha_{0}|0\rangle+\alpha_{1}|1\rangle+\cdots+\alpha_{k-1}|k-1\rangle\), and \(\beta|0\rangle+\) \(\beta|1\rangle+\cdots+\beta|k-1\rangle\), we can compute the inner product of these two vectors, which is \(\sum_{j=0}^{k-1} \alpha_{j}^{*} \beta_{j}\). The absolute value of the inner product is the cosine of the angle between these two vectors in Hilbert space. You should verify that the inner product of any two basis vectors in the standard basis is 0 , showing that they are orthogonal.
The advantage of the ket notation is that the it labels the basis vectors explicitly. This is very convenient because the notation expresses both that the state of the quantum system is a vector, while at the same time explicitly writing out the physical quantity of interest (energy level, position, spin, polarization, etc).