Mathematical Physics Archives - Time Travel, Quantum Entanglement and Quantum Computing https://stationarystates.com/category/mathematical-physics/ Not only is the Universe stranger than we think, it is stranger than we can think...Hiesenberg Wed, 10 Mar 2021 01:44:18 +0000 en-US hourly 1 https://wordpress.org/?v=6.5.3 Examples of Hilbert Spaces https://stationarystates.com/mathematical-physics/examples-of-hilbert-spaces/?utm_source=rss&utm_medium=rss&utm_campaign=examples-of-hilbert-spaces Tue, 09 Feb 2021 13:30:46 +0000 http://stationarystates.com/?p=111 Hilbert Spaces can be FINITE dimensional or INFINITE Dimensional Finite Dimensional Hilbert Spaces n-Tuples of real numbers – – e.g. Inner Product would be just the dot product of two […]

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]]> Hilbert Spaces can be FINITE dimensional or INFINITE Dimensional

Finite Dimensional Hilbert Spaces

n-Tuples of real numbers – R^n – e.g. R^5
Inner Product would be just the dot product of two vectors
\vec{x} = \begin{pmatrix}x1\\x2\\x3\\x4\\x5 \end{pmatrix} and \vec{y} = \begin{bmatrix}y1 & y2 & y3 & y4 & y5 \end{bmatrix}

n-Tuples of complex numbers – C^n – e.g. C^2

\vec{c1} = \begin{pmatrix}a1+ib1 \\a2+ib2 \end{pmatrix}

The inner product here would be the complex inner product: $ (complex  conjugate of c1)^T (c2)

Infinite Dimensional Hilbert Spaces

Space of all complex valued functions with inner product defined in a particular way (square integrable functions)

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]]> Subspaces of Vector Spaces, Invariant Subspaces https://stationarystates.com/mathematical-physics/subspaces-of-vector-spaces/?utm_source=rss&utm_medium=rss&utm_campaign=subspaces-of-vector-spaces Wed, 03 Feb 2021 22:02:55 +0000 http://stationarystates.com/?p=109 The idea is to create a RESTRICTED space within the given vector space. E.g. Within R3 : we can define a space that is far more limited: : This would […]

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]]> The idea is to create a RESTRICTED space within the given vector space.

E.g. Within R3

\R^3: \{ \vec{i}, \vec{j}, \vec{k} \}

we can define a space that is far more limited:

\X^3: \{ \vec{x}, \vec{0}, \vec{-x} \}

This would be a subspace of R3 (it is an R2 space).

An Invariant Subspace

is one that retains the entire subspace under a transformation. In theory, when going from V to V (under T), there is no guarantee where a particular element of W (subset of V) will end up. It could end up anywhere within V. However, if it ends up exactly inside the same subset W, then we say that W is an invariant subspace of V.

An example of an Invariant Transformation Subspace

Consider R3 and consider the transformation of rotating around the z-axis (by an angle theta).

Any point on the X-Y plane will END UP again on the X-Y plane under such a rotation.

We say that the X-Y plane (z=0) is  invariant under the angle rotation transformation.

The connection with Quantum Mechanics?

 

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]]> How to think of complex numbers https://stationarystates.com/mathematical-physics/how-to-think-of-complex-numbers/?utm_source=rss&utm_medium=rss&utm_campaign=how-to-think-of-complex-numbers Fri, 29 Jan 2021 16:35:26 +0000 http://stationarystates.com/?p=93 Think of it as a duo – there’s actually TWO (real) numbers wrapped up in a single complex number. Think of it as a regular vector , but with a […]

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  • Think of it as a duo – there’s actually TWO (real) numbers wrapped up in a single complex number.
  • Think of it as a regular vector , but with a slight twist. When you multiply this vector by a number (real or complex), you end up not just ELONGATING the vector, but also ROTATING it.

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    ]]>         Vector Spaces Examples https://stationarystates.com/mathematical-physics/vector-spaces-examples/?utm_source=rss&utm_medium=rss&utm_campaign=vector-spaces-examples Sun, 24 Jan 2021 03:53:08 +0000 http://stationarystates.com/?p=69 The set of functions that take in a natural number n and return a REAL number.     Functions of ONE Real Variable (i.e. REAL  to REAL)     Of […]

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    ]]>     The set of functions that take in a natural number n and return a REAL number.

        \[ \mathbb{R}^\mathbb{N} = \{f \mid f \colon \mathbb{N} \rightarrow \Re \} \]

    Functions of ONE Real Variable (i.e. REAL  to REAL)

        \[ \mathbb{R}^\mathbb{R} = \{f \mid f \colon \Re \to \Re \} \]

    Of what use are these vector spaces?

    The fundamental theorem (of linear algebra) says that observables (in Quantum Mechanics) are Hermitian Matrices. Which means that the VECTORS form a complete set. That is, the vector space (corresponding to vectors of an observable) is a complete space.

    Note that, this is the fundamental theorem of mathematics, not of Quantum Mechanics. Even though it applies perfectly to Quantum Mechanics. What is ever so mysterious about quantum mechanics is that the language of mathematics can be used throughout, without ever stepping into the physical meaning.

    For instance, the Uncertainty relation is a basic mathematical property of two non-commuting matrices (observables). It has little to do with Quantum Physics, unless we make the connection to satisfy ourselves.

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    ]]>         The nowhere continuous function https://stationarystates.com/mathematical-physics/the-nowhere-continuous-function/?utm_source=rss&utm_medium=rss&utm_campaign=the-nowhere-continuous-function Fri, 22 Jan 2021 18:44:24 +0000 http://stationarystates.com/?p=66 For example, the nowhere continuous function      

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    ]]>     For example, the nowhere continuous function

        \[f(x) = \left\{\begin{matrix}1,~~ x\in \mathbb{Q}\\ 0,~~ x\notin \mathbb{Q}\end{matrix}\right.\]

     

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