Hilbert space

A Hilbert space H is a complete inner product space. It allows the application of linear algebra and calculus techniques in a space that may have an infinite dimension.

The inner product in a Hilbert space has the following properties:

  1. Conjugate symmetry: \langle\phi_1\vert\phi_2\rangle=\langle\phi_2\vert\phi_1\rangle^{*}
  2. Linearity with respect to the 2nd argument: \langle\phi_1\vert c_2\phi_2+c_3\phi_3\rangle=c_2\langle\phi_1\vert \phi_2\rangle+c_3\langle\phi_1\vert \phi_3\rangle
  3. Antilinearity with respect to the first argument: \langle c_1\phi_1+c_2\phi_2\vert \phi_3\rangle=c_1^{*}\langle\phi_1\vert \phi_3\rangle+c_2^{*}\langle\phi_2\vert \phi_3\rangle
  4. Positive semi-definiteness: \langle\phi_1\vert \phi_1\rangle\geq 0, with \langle\phi_1\vert \phi_1\rangle= 0 if \phi_1=0

The last property can be illustrated using the \mathbb{R}^{2} space that is equipped with an inner product. Such a space is an example of a real finite-dimensional Hilbert space. The inner product of the vector \boldsymbol{\mathit{u}} with itself is:

\boldsymbol{\mathit{u}}\cdot\boldsymbol{\mathit{u}}=\begin{pmatrix} u_1 &u_2 \end{pmatrix}\begin{pmatrix} u_1\\u_2 \end{pmatrix}=u{_{1}}^{2}+u{_{2}}^{2}=\{\begin{matrix} >0 &if\; \boldsymbol{\mathit{u}}\neq 0 \\ 0&if \; \boldsymbol{\mathit{u}}=0\end{matrix}

We define a complete Hilbert space as one where every Cauchy sequence in H converges to an element of H. If you recall, a Cauchy sequence is a sequence, e.g. \left \{ x_n \right \}_{n=1}^{\infty} where x_n=\sum_{k=1}^{n}\frac{(-1)^{k+1}}{k}, for which

\lim_{m,n\rightarrow \infty}\left | x_n-x_m \right |=0

We can also define the completeness of a Hilbert space in terms of a sequence of vectors \left \{ \boldsymbol{\mathit{v_n}} \right \}_{n=1}^{\infty}, where \boldsymbol{\mathit{v_n}} =\sum_{k=1}^{n}\boldsymbol{\mathit{u_k}}. Each element \boldsymbol{\mathit{v_n}} is represented by a series of vectors, which converges absolutely (i.e. \sum_{k=1}^{\infty}\left \| \boldsymbol{\mathit{u_k}} \right \|< \infty) and converges to an element of H. In other words, the series of vectors in H converges to some limit vector \boldsymbol{\mathit{L}} in H:

\lim_{n\rightarrow \infty}\left \|\boldsymbol{\mathit{L}}-\sum_{k=1}^{n}\boldsymbol{\mathit{u_k}} \right \|=0

Generally, every element of a vector space can be a point, a vector or a function. In quantum mechanics, we are interested in a Hilbert space called the L^{2} space, where the eigenfunctions of a Hermitian operator are square integrable, i.e. \int_{-a}^{b}\left | \phi(x) \right |^{2}dx< \infty.

Not to be confused with the completeness of a Hilbert space, the completeness of a set of basis eigenfunctions refers to the property that any eigenfunction of the Hilbert space can be expressed as a linear combination of the basis eigenfunctions. An example is the \mathbb{R}^{2} space, where the set of basis vectors \left \{\boldsymbol{\mathit{\hat i}},\boldsymbol{\mathit{\hat j}}\right \} is complete, with a linear combination of \boldsymbol{\mathit{\hat i}} and \boldsymbol{\mathit{\hat j}} spanning \mathbb{R}^{2}. In H, the number of basis vectors \boldsymbol{\mathit{u_k}} may be infinite. If the set of \boldsymbol{\mathit{u_k}} is complete, we say that it spans H, which is itself complete.

Just as the orthonormal vectors \boldsymbol{\mathit{\hat i}} and \boldsymbol{\mathit{\hat j}} form a complete set of basis vectors in the \mathbb{R}^{2} space, where any vector in \mathbb{R}^{2} can be expressed as a linear combination of \boldsymbol{\mathit{\hat i}} and \boldsymbol{\mathit{\hat j}}, we postulate the existence of a complete basis set of orthonormal wavefunctions of any Hermitian operator in L^{2}.



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