Amount square

The graph of the square value function of complex numbers is a paraboloid over the complex number plane

The absolute square or absolute square is a collective term for functions that are mainly used in physics on numbers , vectors and functions . The square of the absolute value of a real or complex number is obtained by squaring its absolute value . The absolute square of a real or complex vector of finite dimension is the square of its length (or Euclidean norm ). The square of the magnitude of a real or complex-valued function is again a function whose function values ​​are equal to the squares of the magnitudes of the functional values ​​of the output function.

The square of the magnitude is used, for example, in signal theory to determine the total energy of a signal. In quantum mechanics , the square of the absolute value is used to calculate the probabilities of states, for example the probabilities of the location of particles. In the theory of relativity , the term absolute square is also used for the Lorentz invariant square of four-vectors in the literature, although this square can also result in negative numbers and thus differs from the general definition in Euclidean spaces .

Definitions

numbers

The graph of the square value function of real numbers is the normal parabola

The square of the absolute value of a real number is simply its square: ${\displaystyle |x|^{2}}$ ${\displaystyle x}$

${\displaystyle |x|^{2}=x^{2}}$.

The square of the absolute value of a complex number with a real part and an imaginary part is (for ) not its square , but: ${\displaystyle |z|^{2}}$ ${\displaystyle z=x+\mathrm {i} y}$ ${\displaystyle \operatorname {Re} (z)=x}$ ${\displaystyle \operatorname {Im} (z)=y}$ ${\displaystyle y\neq 0}$ ${\displaystyle z^{2}=x^{2}+2\mathrm {i} xy-y^{2}}$

${\displaystyle |z|^{2}=z^{\ast }\cdot z=(x-\mathrm {i} y)\cdot (x+\mathrm {i} y)=x^{2}+y^{2}}$.

Here refers to the complex conjugate of . ${\displaystyle z^{\ast }=x-\mathrm {i} y}$ ${\displaystyle z}$

The square of the amount is always a nonnegative real number.

Vectors

In the case of vectors im , the amount or length means the Euclidean norm (2-norm) of the vector. The absolute square of a vector can be calculated with itself using the standard scalar product of the vector: ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle {\vec {v}}\in \mathbb {R} ^{n}}$

${\displaystyle |{\vec {v}}|^{2}={\vec {v}}\cdot {\vec {v}}=v_{1}^{2}+v_{2}^{2}+\dots +v_{n}^{2}}$.

This relationship follows directly from the definition of the Euclidean norm. In the case of complex vectors , the conjugate complex is to be expected accordingly: ${\displaystyle {\vec {v}}\in \mathbb {C} ^{n}}$

${\displaystyle |{\vec {v}}|^{2}=v_{1}^{\ast }\cdot v_{1}+v_{2}^{\ast }\cdot v_{2}+\dots +v_{n}^{\ast }\cdot v_{n}}$.

In both cases the result is a nonnegative real number.

Functions

The square of the sine function

For real or complex-valued functions, the square of the absolute value is defined point by point, which again results in a function. The square of the absolute value of a real-valued function is through ${\displaystyle \phi \colon \Omega \to \mathbb {R} }$

${\displaystyle |\phi |^{2}\colon \Omega \to \mathbb {R} ,~x\mapsto |\phi (x)|^{2}=(\phi (x))^{2}}$

given and thus equal to the square of the function, while the square of the magnitude of a complex-valued function is given by ${\displaystyle \phi \colon \Omega \to \mathbb {C} }$

${\displaystyle |\phi |^{2}\colon \Omega \to \mathbb {C} ,~x\mapsto |\phi (x)|^{2}=(\phi (x))^{\ast }\cdot \phi (x)}$

is defined. The square of the absolute value of a function is therefore a real-valued function with the same definition range , the functional values ​​of which are equal to the square of the absolute values ​​of the functional values ​​of the output function. In the real case it is also notated through and in the complex case also through . ${\displaystyle \Omega }$ ${\displaystyle \phi ^{2}}$ ${\displaystyle \phi ^{\ast }\phi }$

properties

The following are basic properties of the square magnitude of complex numbers. By looking at them point by point, these properties can also be transferred to functions. Properties of the absolute square of vectors can be found in the article Euclidean Norm .

Reciprocal

The following applies to the reciprocal of a complex number ${\displaystyle z\neq 0}$

${\displaystyle {\frac {1}{z}}={\frac {z^{*}}{z^{*}z}}={\frac {z^{*}}{|z|^{2}}}}$.

It can therefore be calculated by dividing the complex conjugate number by the square of the absolute value.

Amount of square

The square of the magnitude of a complex number is equal to the magnitude of the square of the number, that is

${\displaystyle |z|^{2}=|z^{2}|}$.

It is true

${\displaystyle |z^{2}|=|(x+\mathrm {i} y)^{2}|=|x^{2}-y^{2}+2\mathrm {i} xy|={\sqrt {(x^{2}-y^{2})^{2}+(2xy)^{2}}}={\sqrt {(x^{2}+y^{2})^{2}}}=x^{2}+y^{2}=|z|^{2}}$.

When displaying in polar form with , one receives accordingly ${\displaystyle z=r\cdot \mathrm {e} ^{\mathrm {i} \varphi }}$ ${\displaystyle r=|z|}$

${\displaystyle |z^{2}|=|r^{2}\mathrm {e} ^{2\mathrm {i} \varphi }|=|r^{2}|\cdot |\mathrm {e} ^{2\mathrm {i} \varphi }|=r^{2}\cdot 1=|z|^{2}}$.

Product and quotient

For the square of the amount of the product of two complex numbers and the following applies: ${\displaystyle z_{1}=r_{1}\mathrm {e} ^{\mathrm {i} \varphi _{1}}}$ ${\displaystyle z_{2}=r_{2}\mathrm {e} ^{\mathrm {i} \varphi _{2}}}$

${\displaystyle |z_{1}z_{2}|^{2}=(r_{1}\mathrm {e} ^{-\mathrm {i} \varphi _{1}}r_{2}\mathrm {e} ^{-\mathrm {i} \varphi _{2}})(r_{1}\mathrm {e} ^{\mathrm {i} \varphi _{1}}r_{2}\mathrm {e} ^{\mathrm {i} \varphi _{2}})=r_{1}^{2}r_{2}^{2}=|z_{1}|^{2}|z_{2}|^{2}}$.

Similarly, for the square of the magnitude of the quotient of two complex numbers, the following applies : ${\displaystyle z_{2}\neq 0}$

${\displaystyle \left|{\frac {z_{1}}{z_{2}}}\right|^{2}=\left({\frac {r_{1}\mathrm {e} ^{-\mathrm {i} \varphi _{1}}}{r_{2}\mathrm {e} ^{-\mathrm {i} \varphi _{2}}}}\right)\left({\frac {r_{1}\mathrm {e} ^{\mathrm {i} \varphi _{1}}}{r_{2}\mathrm {e} ^{\mathrm {i} \varphi _{2}}}}\right)={\frac {r_{1}^{2}}{r_{2}^{2}}}={\frac {|z_{1}|^{2}}{|z_{2}|^{2}}}}$.

The square of the amount of the product or the quotient of two complex numbers is therefore the product or the quotient of their squared amounts. The amount itself already has these properties.

Sum and difference

The following applies to the square of the sum or the difference between two complex numbers:

${\displaystyle |z_{1}\pm z_{2}|^{2}=(r_{1}\mathrm {e} ^{-\mathrm {i} \varphi _{1}}\pm r_{2}\mathrm {e} ^{-\mathrm {i} \varphi _{2}})(r_{1}\mathrm {e} ^{\mathrm {i} \varphi _{1}}\pm r_{2}\mathrm {e} ^{\mathrm {i} \varphi _{2}})=r_{1}^{2}+r_{2}^{2}\pm 2r_{1}r_{2}\cos(\varphi _{2}-\varphi _{1})}$.

If one imagines the complex numbers and their sum or difference as points in the complex plane, then this relationship corresponds precisely to the cosine law for the resulting triangle. Specifically, one obtains for the square of the sum of two complex numbers with the amount one: ${\displaystyle z_{1}}$ ${\displaystyle z_{2}}$ ${\displaystyle z_{1}\pm z_{2}}$

${\displaystyle |\mathrm {e} ^{\mathrm {i} \varphi _{1}}+\mathrm {e} ^{\mathrm {i} \varphi _{2}}|^{2}=2+2\cos(\varphi _{2}-\varphi _{1})=4\cos ^{2}\left({\frac {\varphi _{2}-\varphi _{1}}{2}}\right)}$.

Applications

Signal theory

In signal theory , the total energy or the total power of a continuous complex-valued signal is defined as the integral over its square of magnitude, that is ${\displaystyle f\colon \mathbb {R} \to \mathbb {C} }$

${\displaystyle E=\int _{-\infty }^{\infty }|f(t)|^{2}\,\mathrm {d} t}$.

The total energy thus corresponds to the square of the standard of the signal. A central result here is Plancherel's theorem , according to which the energy of a signal in the time domain is equal to its energy in the frequency domain. If then the (normalized) Fourier transform of , then applies ${\displaystyle L^{2}}$ ${\displaystyle {\mathcal {F}}}$ ${\displaystyle f}$

${\displaystyle \int _{-\infty }^{\infty }|f(t)|^{2}\,\mathrm {d} t=\int _{-\infty }^{\infty }|{\mathcal {F}}(\omega )|^{2}\,\mathrm {d} \omega }$.

The Fourier transformation thus receives the total energy of a signal and thus represents a unitary mapping .

theory of relativity

In the theory of relativity , the time and location coordinates of an event in space-time are summarized in a four-vector position . The time coordinate is multiplied by the speed of light so that, like the spatial coordinates, it has the dimension of a length. In the Minkowski space of flat spacetime - in contrast to the definition given above for vectors in - the square of the four-vector becomes through ${\displaystyle r=(c\,t,x,y,z)}$ ${\displaystyle t}$ ${\displaystyle c}$ ${\displaystyle (x,y,z)}$ ${\displaystyle \mathbb {R} ^{4}}$ ${\displaystyle r}$

${\displaystyle r^{2}=c^{2}\,t^{2}-x^{2}-y^{2}-z^{2}}$

defines what can also result in a negative real number. The term absolute square is also used in the literature for this four-vector square, although the bilinear form defined on the Minkowski space that induces this square is not a scalar product from which a square with non-negative values ​​in the above sense can be derived. The Lorentz transformations can now be characterized as those coordinate transformations which contain the bilinear form and thus the square of the absolute value. For example, the coordinate transformation into the rest system of an object moving with relative speed in the direction ${\displaystyle v}$ ${\displaystyle x}$

${\displaystyle t'=\gamma (t-vx/c^{2}),\quad x'=\gamma (x-vt),\quad y'=y,\quad z'=z}$,

whereby the Lorentz factor is length-preserving, that is, for the four-vector transformed applies ${\displaystyle \gamma }$ ${\displaystyle r'=(c\,t',x',y',z')}$

${\displaystyle (r')^{2}=r^{2}}$.

Analogously to this, the square of the absolute value of every other four-vector (for example the momentum four-vector ) is defined, which is then also invariant with respect to a Lorentz transformation.

Quantum mechanics

The square of the absolute value is also frequently used in quantum mechanics. In Bra-Ket notation, the scalar product of two vectors and the underlying Hilbert space is written as. Is an observable given as an operator with a non-degenerate eigenvalue to a normalized eigenvector , that is ${\displaystyle |\phi \rangle }$ ${\displaystyle |\psi \rangle }$ ${\displaystyle \langle \phi |\psi \rangle }$ ${\displaystyle A}$ ${\displaystyle a}$ ${\displaystyle |u\rangle }$

${\displaystyle A|u\rangle =a|u\rangle }$,

the probability of measuring the value for the observable in a state is calculated using the square of the magnitude of the corresponding probability amplitude: ${\displaystyle |\psi \rangle }$ ${\displaystyle a}$ ${\displaystyle A}$

${\displaystyle {\mathcal {P}}(a)=\left|\langle u|\psi \rangle \right|^{2}}$.

The absolute square in the point-wise sense of the normalized wave function from the Schrödinger equation is equal to the probability density of the particle:

${\displaystyle \rho (\mathbf {r} ,t)=|\psi (\mathbf {r} ,t)|^{2}=\psi ^{\ast }(\mathbf {r} ,t)\,\psi (\mathbf {r} ,t)}$.

algebra

In body theory , the square of the absolute value of complex numbers is the norm of body expansion . It also represents the norm in the quadratic number field and therefore plays an important role when calculating with Gaussian numbers . ${\displaystyle \mathbb {C} /\mathbb {R} }$ ${\displaystyle \mathbb {Q} (i)}$

Individual evidence

1. ↑ May-Britt Kallenrode: Calculation methods of physics: Mathematical companion for experimental physics . Springer, 2005, ISBN 3-540-27482-0 , pp. 91 .
2. ↑ Klemens Burg, Herbert Haf, Friedrich Wille, Andreas Meister: Higher Mathematics For Engineers Volume II: Linear Algebra . Springer, 2012, ISBN 3-8348-2267-1 , pp. 46 .
3. ^ Klaus Stierstadt: Thermodynamics: From the microphysics to the macrophysics . Springer, 2010, ISBN 3-642-05098-0 , pp. 83-84 .
4. ^ Ilja N. Bronstein, Konstantin A. Semendjajew, Gerhard Musiol, Heiner Mühlig: Taschenbuch der Mathematik . 7th edition. Harri Deutsch, 2008, ISBN 3-8171-2007-9 , pp. 37 .
5. ^ ^{A b} Eric W. Weisstein: CRC Concise Encyclopedia of Mathematics . 2nd Edition. CRC Press, 2010, ISBN 1-4200-3522-3 , pp. 22 .
6. ↑ Uwe Kiencke, Michael Schwarz, Thomas Weickert: Signal processing: time-frequency analysis and estimation methods . Oldenbourg, 2008, ISBN 3-486-58668-8 , pp. 401 .
7. ↑ Peter Burger, Ute Diemar, Eberhard Kallenbach, Bernd Marx, Tom Ströhla: Theoretical Foundations of Electrical Engineering 2 . Springer, 2006, ISBN 3-519-00525-5 , pp. 103-104 .
8. ^ Claude Cohen-Tannoudji, Bernard Diu, Franck Laloë: Quantum Mechanics, Volume 1 . 3. Edition. de Gruyter, 2007, ISBN 3-11-019324-8 , p. 90 f .

Web links

Wiktionary: amount square  - explanations of meanings, word origins, synonyms, translations

• Eric W. Weisstein : Absolute Square . In: MathWorld (English).