diff options
Diffstat (limited to 'source/know/concept')
-rw-r--r-- | source/know/concept/capillary-action/index.md | 126 | ||||
-rw-r--r-- | source/know/concept/capillary-length/index.md | 79 | ||||
-rw-r--r-- | source/know/concept/cartesian-coordinates/index.md | 4 | ||||
-rw-r--r-- | source/know/concept/drude-model/index.md | 6 | ||||
-rw-r--r-- | source/know/concept/jurins-law/index.md | 78 | ||||
-rw-r--r-- | source/know/concept/orthogonal-curvilinear-coordinates/index.md | 6 | ||||
-rw-r--r-- | source/know/concept/parabolic-cylindrical-coordinates/index.md (renamed from source/know/concept/cylindrical-parabolic-coordinates/index.md) | 12 | ||||
-rw-r--r-- | source/know/concept/polar-cylindrical-coordinates/index.md (renamed from source/know/concept/cylindrical-polar-coordinates/index.md) | 8 | ||||
-rw-r--r-- | source/know/concept/sturm-liouville-theory/index.md | 321 |
9 files changed, 339 insertions, 301 deletions
diff --git a/source/know/concept/capillary-action/index.md b/source/know/concept/capillary-action/index.md deleted file mode 100644 index fea6ef8..0000000 --- a/source/know/concept/capillary-action/index.md +++ /dev/null @@ -1,126 +0,0 @@ ---- -title: "Capillary action" -sort_title: "Capillary action" -date: 2021-03-29 -categories: -- Physics -- Fluid mechanics -- Fluid statics -- Surface tension -layout: "concept" ---- - -**Capillary action** refers to the movement of liquid -through narrow spaces due to surface tension, often against gravity. -It occurs when the [Laplace pressure](/know/concept/young-laplace-law/) -from surface tension is much larger in magnitude than the -[hydrostatic pressure](/know/concept/hydrostatic-pressure/) from gravity. - -Consider a spherical droplet of liquid with radius $$R$$. -The hydrostatic pressure difference -between the top and bottom of the drop -is much smaller than the Laplace pressure: - -$$\begin{aligned} - 2 R \rho g \ll 2 \frac{\alpha}{R} -\end{aligned}$$ - -Where $$\rho$$ is the density of the liquid, -$$g$$ is the acceleration due to gravity, -and $$\alpha$$ is the energy cost per unit surface area. -Rearranging the inequality yields: - -$$\begin{aligned} - R^2 \ll \frac{\alpha}{\rho g} -\end{aligned}$$ - -From the right-hand side we define the **capillary length** $$L_c$$, -so gravity is negligible if $$R \ll L_c$$: - -$$\begin{aligned} - \boxed{ - L_c - \equiv \sqrt{\frac{\alpha}{\rho g}} - } -\end{aligned}$$ - -In general, for a system with characteristic length $$L$$, -the relative strength of gravity compared to surface tension -is described by the **Bond number** $$\mathrm{Bo}$$ -or **Eötvös number** $$\mathrm{Eo}$$: - -$$\begin{aligned} - \boxed{ - \mathrm{Bo} - \equiv \mathrm{Eo} - \equiv \frac{L^2}{L_c^2} - = \frac{m g}{\alpha L} - } -\end{aligned}$$ - -The rightmost side gives an alternative way of understanding $$\mathrm{Bo}$$: -$$m$$ is the mass of a cube with side $$L$$, such that the numerator is the weight force, -and the denominator is the tension force of the surface. -In any case, capillary action can be observed when $$\mathrm{Bo \ll 1}$$. - -The most famous example of capillary action is **capillary rise**, -where a liquid "climbs" upwards in a narrow vertical tube with radius $$R$$, -apparently defying gravity. -Assuming the liquid-air interface is a spherical cap -with constant [curvature](/know/concept/curvature/) radius $$R_c$$, -then we know that the liquid is at rest -when the hydrostatic pressure equals the Laplace pressure: - -$$\begin{aligned} - \rho g h - \approx \alpha \frac{2}{R_c} - = 2 \alpha \frac{\cos\theta}{R} -\end{aligned}$$ - -Where $$\theta$$ is the liquid-tube contact angle, -and we are neglecting variations of the height $$h$$ due to the curvature -(i.e. the [meniscus](/know/concept/meniscus/)). -By isolating the above equation for $$h$$, -we arrive at **Jurin's law**, -which predicts the height climbed by a liquid in a tube with radius $$R$$: - -$$\begin{aligned} - \boxed{ - h - = 2 \frac{L_c^2}{R} \cos\theta - } -\end{aligned}$$ - -Depending on $$\theta$$, $$h$$ can be negative, -i.e. the liquid might descend below the ambient level. - - -An alternative derivation of Jurin's law balances the forces instead of the pressures. -On the right, we have the gravitational force -(i.e. the energy-per-distance to lift the liquid), -and on the left, the surface tension force -(i.e. the energy-per-distance of the liquid-tube interface): - -$$\begin{aligned} - \pi R^2 \rho g h - \approx 2 \pi R (\alpha_{sg} - \alpha_{sl}) -\end{aligned}$$ - -Where $$\alpha_{sg}$$ and $$\alpha_{sl}$$ are the energy costs -of the solid-gas and solid-liquid interfaces. -Thanks to the [Young-Dupré relation](/know/concept/young-dupre-relation/), -we can rewrite this as follows: - -$$\begin{aligned} - R \rho g h - = 2 \alpha \cos\theta -\end{aligned}$$ - -Isolating this for $$h$$ simply yields Jurin's law again, as expected. - - - -## References -1. B. Lautrup, - *Physics of continuous matter: exotic and everyday phenomena in the macroscopic world*, 2nd edition, - CRC Press. diff --git a/source/know/concept/capillary-length/index.md b/source/know/concept/capillary-length/index.md new file mode 100644 index 0000000..4dbb788 --- /dev/null +++ b/source/know/concept/capillary-length/index.md @@ -0,0 +1,79 @@ +--- +title: "Capillary length" +sort_title: "Capillary length" +date: 2021-03-29 +categories: +- Physics +- Fluid mechanics +- Fluid statics +- Surface tension +layout: "concept" +--- + +**Capillary action** refers to the movement of liquid +through narrow spaces due to surface tension, often against gravity. +It occurs when the [Laplace pressure](/know/concept/young-laplace-law/) +from surface tension is much larger in magnitude than the +[hydrostatic pressure](/know/concept/hydrostatic-pressure/) from gravity. + +Consider a spherical droplet of liquid with radius $$R$$. +The hydrostatic pressure difference +between the top and bottom of the drop +is much smaller than the Laplace pressure: + +$$\begin{aligned} + 2 R \rho g \ll 2 \frac{\alpha}{R} +\end{aligned}$$ + +Where $$\rho$$ is the density of the liquid, +$$g$$ is the acceleration due to gravity, +and $$\alpha$$ is the energy cost per unit surface area. +Rearranging the inequality yields: + +$$\begin{aligned} + R^2 \ll \frac{\alpha}{\rho g} +\end{aligned}$$ + +From this, we define the **capillary length** $$L_c$$ +such that gravity is negligible if $$R \ll L_c$$: + +$$\begin{aligned} + \boxed{ + L_c + \equiv \sqrt{\frac{\alpha}{\rho g}} + } +\end{aligned}$$ + +In general, for a system with characteristic length $$L$$, +the relative strength of gravity compared to surface tension +is described by the **Bond number** $$\mathrm{Bo}$$ +or **Eötvös number** $$\mathrm{Eo}$$: + +$$\begin{aligned} + \boxed{ + \mathrm{Bo} + \equiv \mathrm{Eo} + \equiv \frac{L^2}{L_c^2} + } +\end{aligned}$$ + +Capillary action is observed when $$\mathrm{Bo \ll 1}$$, +while for $$\mathrm{Bo} \gg 1$$ surface tension is negligible. + +For an alternative interpretation of $$\mathrm{Bo}$$, +let $$m \equiv \rho L^3$$ be the mass of a cube with side $$L$$ +such that its weight is $$m g$$. +The tension force on its face is $$\alpha L$$, +so $$\mathrm{Bo}$$ is simply the force ratio: + +$$\begin{aligned} + \mathrm{Bo} + = \frac{m g}{\alpha L} +\end{aligned}$$ + + + +## References +1. B. Lautrup, + *Physics of continuous matter: exotic and everyday phenomena in the macroscopic world*, 2nd edition, + CRC Press. diff --git a/source/know/concept/cartesian-coordinates/index.md b/source/know/concept/cartesian-coordinates/index.md index d198e84..a0bfc39 100644 --- a/source/know/concept/cartesian-coordinates/index.md +++ b/source/know/concept/cartesian-coordinates/index.md @@ -11,8 +11,8 @@ layout: "concept" This article is a supplement to the ones on [orthogonal curvilinear systems](/know/concept/orthogonal-curvilinear-coordinates/), [spherical coordinates](/know/concept/spherical-coordinates/), -[cylindrical polar coordinates](/know/concept/cylindrical-polar-coordinates/), -and [cylindrical parabolic coordinates](/know/concept/cylindrical-parabolic-coordinates/). +[polar cylindrical coordinates](/know/concept/polar-cylindrical-coordinates/), +and [parabolic cylindrical coordinates](/know/concept/parabolic-cylindrical-coordinates/). The well-known Cartesian coordinate system $$(x, y, z)$$ has trivial **scale factors**: diff --git a/source/know/concept/drude-model/index.md b/source/know/concept/drude-model/index.md index c4faf81..0026d90 100644 --- a/source/know/concept/drude-model/index.md +++ b/source/know/concept/drude-model/index.md @@ -11,7 +11,7 @@ layout: "concept" The **Drude model**, also known as the **Drude-Lorentz model** due to its analogy -to the *Lorentz oscillator model* +to the [Lorentz oscillator model](/know/concept/lorentz-oscillator-model/) classically predicts the [dielectric function](/know/concept/dielectric-function/) and electric conductivity of a gas of free charges, as found in metals and doped semiconductors. @@ -59,7 +59,7 @@ $$\begin{aligned} = - \frac{N q^2}{m (\omega^2 + i \gamma \omega)} \vb{E}(t) \end{aligned}$$ -The electric displacement field $$\vb{D}$$ is then as follows, +The electric displacement field $$\vb{D}(t)$$ is then as follows, where the parenthesized expression is the dielectric function $$\varepsilon_r$$ of the material: @@ -180,7 +180,7 @@ We must replace the carriers' true mass $$m$$ with their *effective mass* $$m^*$ found from the material's electronic band structure. Furthermore, semiconductors already have a high intrinsic dielectric function $$\varepsilon_{\mathrm{int}}$$ -before being doped, so the displacement field $$\vb{D}$$ becomes: +before being doped, so the displacement field $$\vb{D}(t)$$ becomes: $$\begin{aligned} \vb{D} diff --git a/source/know/concept/jurins-law/index.md b/source/know/concept/jurins-law/index.md new file mode 100644 index 0000000..6214477 --- /dev/null +++ b/source/know/concept/jurins-law/index.md @@ -0,0 +1,78 @@ +--- +title: "Jurin's law" +sort_title: "Jurin's law" +date: 2023-06-15 +categories: +- Physics +- Fluid mechanics +- Fluid statics +- Surface tension +layout: "concept" +--- + +A well-known example of *capillary action* is +when a liquid climbs up a narrow vertical tube with radius $$R$$, +apparently defying gravity. +Indeed, this occurs when the liquid's surface tension can overpower gravity; +specifically, when the [capillary length](/know/concept/capillary-length/) $$L_c > R$$. + +Let us assume that the liquid-air interface has a spherical shape, +which may point up or down depending on the liquid. +This interface then has a constant [curvature radius](/know/concept/curvature/) $$r$$ +determined by the contact angle $$\theta$$ of the liquid to the tube: +$$r = R / \cos{\theta}$$. We know that the liquid is at rest +when the [hydrostatic pressure](/know/concept/hydrostatic-pressure/) +equals the resulting [Laplace pressure](/know/concept/young-laplace-law/): + +$$\begin{aligned} + \rho g h + = \alpha \frac{2}{r} + = 2 \alpha \frac{\cos{\theta}}{R} +\end{aligned}$$ + +Note that $$h$$ is the height of interface's highest/lowest point; +we neglect the [meniscus](/know/concept/meniscus/). +By isolating the above equation for $$h$$, we arrive at **Jurin's law**: + +$$\begin{aligned} + \boxed{ + h + = \frac{2 \alpha \cos{\theta}}{\rho g R} + = 2 \frac{L_c^2}{R} \cos{\theta} + } +\end{aligned}$$ + +Where $$L_c \equiv \sqrt{\alpha / \rho g}$$. +This predicts the height climbed by a liquid in a narrow tube. +If $$\theta > 90\degree$$, then $$h$$ is negative, +i.e. the liquid descends below the ambient level. + +An alternative derivation of Jurin's law balances the forces instead of the pressures. +On the right, we have the gravitational force +(i.e. the energy-per-distance to lift the liquid), +and on the left, the surface tension force +(i.e. the energy-per-distance of the liquid-tube interface): + +$$\begin{aligned} + \pi R^2 \rho g h + \approx 2 \pi R (\alpha_{sg} - \alpha_{sl}) +\end{aligned}$$ + +Where $$\alpha_{sg}$$ and $$\alpha_{sl}$$ are the energy costs +of the solid-gas and solid-liquid interfaces. +Thanks to the [Young-Dupré relation](/know/concept/young-dupre-relation/), +we can rewrite this as follows: + +$$\begin{aligned} + R \rho g h + = 2 \alpha \cos\theta +\end{aligned}$$ + +Isolating this for $$h$$ simply yields Jurin's law again, as expected. + + + +## References +1. B. Lautrup, + *Physics of continuous matter: exotic and everyday phenomena in the macroscopic world*, 2nd edition, + CRC Press. diff --git a/source/know/concept/orthogonal-curvilinear-coordinates/index.md b/source/know/concept/orthogonal-curvilinear-coordinates/index.md index 669358c..4714a95 100644 --- a/source/know/concept/orthogonal-curvilinear-coordinates/index.md +++ b/source/know/concept/orthogonal-curvilinear-coordinates/index.md @@ -18,10 +18,10 @@ where at least one of the coordinate surfaces is curved: e.g. in cylindrical coordinates, the coordinate line of $$r$$ and $$z$$ is a circle. Here we limit ourselves to **orthogonal** systems, where the coordinate surfaces are always perpendicular. -Examples of such orthogonal curvilinear systems include +Examples of such orthogonal curvilinear systems are [spherical coordinates](/know/concept/spherical-coordinates/), -[cylindrical polar coordinates](/know/concept/cylindrical-polar-coordinates/), -[cylindrical parabolic coordinates](/know/concept/cylindrical-parabolic-coordinates/), +[polar cylindrical coordinates](/know/concept/polar-cylindrical-coordinates/), +[parabolic cylindrical coordinates](/know/concept/parabolic-cylindrical-coordinates/), and (trivially) [Cartesian coordinates](/know/concept/cartesian-coordinates/). diff --git a/source/know/concept/cylindrical-parabolic-coordinates/index.md b/source/know/concept/parabolic-cylindrical-coordinates/index.md index 58358dd..6ba19f5 100644 --- a/source/know/concept/cylindrical-parabolic-coordinates/index.md +++ b/source/know/concept/parabolic-cylindrical-coordinates/index.md @@ -1,6 +1,6 @@ --- -title: "Cylindrical parabolic coordinates" -sort_title: "Cylindrical parabolic coordinates" +title: "Parabolic cylindrical coordinates" +sort_title: "Parabolic cylindrical coordinates" date: 2021-03-04 categories: - Mathematics @@ -8,7 +8,7 @@ categories: layout: "concept" --- -**Cylindrical parabolic coordinates** extend parabolic coordinates $$(\sigma, \tau)$$ to 3D, +**Parabolic cylindrical coordinates** extend parabolic coordinates $$(\sigma, \tau)$$ to 3D, by describing a point in space using the variables $$(\sigma, \tau, z)$$. The $$z$$-axis is the same as in the Cartesian system, (hence the name *cylindrical*), while the coordinate lines of $$\sigma$$ and $$\tau$$ are confocal parabolas. @@ -38,10 +38,10 @@ $$\begin{aligned} \boxed{ \begin{aligned} \sigma - &= \sgn(x) \sqrt{\sqrt{x^2 + y^2} - x} + &= \sqrt{\sqrt{x^2 + y^2} - x} \\ \tau - &= \sqrt{\sqrt{x^2 + y^2} + x} + &= \sgn(y) \sqrt{\sqrt{x^2 + y^2} + x} \\ z &= z @@ -49,7 +49,7 @@ $$\begin{aligned} } \end{aligned}$$ -Cylindrical parabolic coordinates form +Parabolic cylindrical coordinates form an [orthogonal curvilinear system](/know/concept/orthogonal-curvilinear-coordinates/), whose **scale factors** $$h_\sigma$$, $$h_\tau$$ and $$h_z$$ we need. To get those, we calculate the unnormalized local basis: diff --git a/source/know/concept/cylindrical-polar-coordinates/index.md b/source/know/concept/polar-cylindrical-coordinates/index.md index cf227a6..2223996 100644 --- a/source/know/concept/cylindrical-polar-coordinates/index.md +++ b/source/know/concept/polar-cylindrical-coordinates/index.md @@ -1,6 +1,6 @@ --- -title: "Cylindrical polar coordinates" -sort_title: "Cylindrical polar coordinates" +title: "Polar cylindrical coordinates" +sort_title: "Polar cylindrical coordinates" date: 2021-07-26 categories: - Mathematics @@ -8,7 +8,7 @@ categories: layout: "concept" --- -**Cylindrical polar coordinates** extend polar coordinates $$(r, \varphi)$$ to 3D, +**Polar cylindrical coordinates** extend polar coordinates $$(r, \varphi)$$ to 3D, by describing the location of a point in space using the variables $$(r, \varphi, z)$$. The $$z$$-axis is unchanged from the Cartesian system, @@ -52,7 +52,7 @@ $$\begin{aligned} } \end{aligned}$$ -Cylindrical polar coordinates form +Polar cylindrical coordinates form an [orthogonal curvilinear system](/know/concept/orthogonal-curvilinear-coordinates/), whose **scale factors** $$h_r$$, $$h_\varphi$$ and $$h_z$$ we need. To get those, we calculate the unnormalized local basis: diff --git a/source/know/concept/sturm-liouville-theory/index.md b/source/know/concept/sturm-liouville-theory/index.md index bff57af..d7984b5 100644 --- a/source/know/concept/sturm-liouville-theory/index.md +++ b/source/know/concept/sturm-liouville-theory/index.md @@ -8,14 +8,15 @@ categories: layout: "concept" --- -**Sturm-Liouville theory** defines the analogue of Hermitian matrix -eigenvalue problems for linear second-order ODEs. +**Sturm-Liouville theory** extends +the concept of Hermitian matrix eigenvalue problems +to linear second-order ordinary differential equations. -It states that, given suitable boundary conditions, any linear -second-order ODE can be rewritten using the **Sturm-Liouville operator**, -and that the corresponding eigenvalue problem, known as a -**Sturm-Liouville problem**, will give real eigenvalues and a complete set -of eigenfunctions. +It states that, given suitable boundary conditions, +any such equation can be rewritten using the **Sturm-Liouville operator**, +and that the corresponding eigenvalue problem, +known as a **Sturm-Liouville problem**, +will give real eigenvalues and a complete set of eigenfunctions. @@ -23,18 +24,19 @@ of eigenfunctions. Consider the most general form of a second-order linear differential operator $$\hat{L}$$, where $$p_0(x)$$, $$p_1(x)$$, and $$p_2(x)$$ -are real functions of $$x \in [a,b]$$ which are nonzero for all $$x \in ]a, b[$$: +are real functions of $$x \in [a,b]$$ and are nonzero for all $$x \in \,\,]a, b[$$: $$\begin{aligned} - \hat{L} \{u(x)\} = p_0(x) u''(x) + p_1(x) u'(x) + p_2(x) u(x) + \hat{L} \{u(x)\} + \equiv p_2(x) \: u''(x) + p_1(x) \: u'(x) + p_0(x) \: u(x) \end{aligned}$$ -We now define the **adjoint** or **Hermitian** operator -$$\hat{L}^\dagger$$ analogously to matrices: +Analogously to matrices, +we now define its **adjoint** operator $$\hat{L}^\dagger$$ as follows: $$\begin{aligned} - \inprod{f}{\hat{L} g} - = \inprod{\hat{L}^\dagger f}{g} + \inprod{\hat{L}^\dagger f}{g} + \equiv \inprod{f}{\hat{L} g} \end{aligned}$$ What is $$\hat{L}^\dagger$$, given the above definition of $$\hat{L}$$? @@ -43,146 +45,155 @@ We start from the inner product $$\inprod{f}{\hat{L} g}$$: $$\begin{aligned} \inprod{f}{\hat{L} g} &= \int_a^b f^*(x) \hat{L}\{g(x)\} \dd{x} - = \int_a^b (f^* p_0) g'' + (f^* p_1) g' + (f^* p_2) g \dd{x} + = \int_a^b (f^* p_2) g'' + (f^* p_1) g' + (f^* p_0) g \dd{x} \\ - &= \big[ (f^* p_0) g' + (f^* p_1) g \big]_a^b - \int_a^b (f^* p_0)' g' + (f^* p_1)' g - (f^* p_2) g \dd{x} + &= \Big[ (f^* p_2) g' + (f^* p_1) g \Big]_a^b - \int_a^b (f^* p_2)' g' + (f^* p_1)' g - (f^* p_0) g \dd{x} \\ - &= \big[ f^* \big( p_0 g' \!+\! p_1 g \big) \!-\! (f^* p_0)' g \big]_a^b + \int_a^b \! \big( (f p_0)'' - (f p_1)' + (f p_2) \big)^* g \dd{x} + &= \Big[ f^* (p_2 g' + p_1 g) - (f^* p_2)' g \Big]_a^b + \int_a^b \! \Big( (f p_2)'' - (f p_1)' + (f p_0) \Big)^* g \dd{x} \\ - &= \big[ f^* \big( p_0 g' + (p_1 - p_0') g \big) - (f^*)' p_0 g \big]_a^b + \int_a^b \big( \hat{L}^\dagger\{f\} \big)^* g \dd{x} -\end{aligned}$$ - -We now have an expression for $$\hat{L}^\dagger$$, but are left with an -annoying boundary term: - -$$\begin{aligned} - \inprod{f}{\hat{L} g} - &= \big[ f^* \big( p_0 g' + (p_1 - p_0') g \big) - (f^*)' p_0 g \big]_a^b + \inprod{\hat{L}^\dagger f}{g} + &= \Big[ f^* \big( p_2 g' + (p_1 - p_2') g \big) - (f^*)' p_2 g \Big]_a^b + \int_a^b \Big( \hat{L}^\dagger\{f\} \Big)^* g \dd{x} \end{aligned}$$ -To fix this, -let us demand that $$p_1(x) = p_0'(x)$$ and that -$$[p_0(f^* g' - (f^*)' g)]_a^b = 0$$, leaving: +The newly-formed operator on $$f$$ must be $$\hat{L}^\dagger$$, +but there is an additional boundary term. +To fix this, we demand that $$p_1(x) = p_2'(x)$$ +and that $$\big[ p_2 (f^* g' - (f^*)' g) \big]_a^b = 0$$, leaving: $$\begin{aligned} \inprod{f}{\hat{L} g} - &= \big[ p_0 \big( f^* g' - (f^*)' g \big) \big]_a^b + \inprod{\hat{L}^\dagger f}{g} - = \inprod{\hat{L}^\dagger f}{g} + &= \Big[ f^* \big( p_2 g' + (p_1 - p_2') g \big) - (f^*)' p_2 g \Big]_a^b + \inprod{\hat{L}^\dagger f}{g} + \\ + &= \Big[ p_2 \big( f^* g' - (f^*)' g \big) \Big]_a^b + \inprod{\hat{L}^\dagger f}{g} + \\ + &= \inprod{\hat{L}^\dagger f}{g} \end{aligned}$$ -Using the aforementioned restriction $$p_1(x) = p_0'(x)$$, -we then take a look at the definition of $$\hat{L}^\dagger$$: +Let us look at the expression for $$\hat{L}^\dagger$$ we just found, +with the restriction $$p_1 = p_2'$$ in mind: $$\begin{aligned} \hat{L}^\dagger \{f\} - &= (p_0 f)'' - (p_1 f)' + (p_2 f) + &= (p_2 f)'' - (p_1 f)' + (p_0 f) \\ - &= p_0 f'' + (2 p_0' - p_1) f' + (p_0'' - p_1' + p_2) f + &= (p_2'' f + 2 p_2' f' + p_2 f'') - (p_1' f + p_1 f') + (p_0 f) \\ - &= p_0 f'' + p_0' f' + p_2 f + &= p_2 f'' + (2 p_2' - p_1) f' + (p_2'' - p_1' + p_0) f \\ - &= (p_0 f')' + p_2 f + &= p_2 f'' + p_1 f' + p_0 f + \\ + &= \hat{L}\{f\} \end{aligned}$$ -The original operator $$\hat{L}$$ reduces to the same form, -so it is **self-adjoint**: +So $$\hat{L}$$ is **self-adjoint**, i.e. $$\hat{L}^\dagger$$ is the same as $$\hat{L}$$! +Indeed, every such second-order linear operator is self-adjoint +if it satisfies the constraints $$p_1 = p_2'$$ and $$\big[ p_2 (f^* g' - (f^*)' g) \big]_a^b = 0$$. + +But what if $$p_1 \neq p_2'$$? +Let us multiply $$\hat{L}$$ by an unknown $$p(x) \neq 0$$ +and divide by $$p_2(x) \neq 0$$: $$\begin{aligned} - \hat{L} \{f\} - &= p_0 f'' + p_0' f' + p_2 f - = (p_0 f')' + p_2 f - = \hat{L}^\dagger \{f\} + \frac{p}{p_2} \hat{L} \{u\} + = p u'' + p \frac{p_1}{p_2} u' + p \frac{p_0}{p_2} u \end{aligned}$$ -Consequently, every such second-order linear operator $$\hat{L}$$ is self-adjoint, -as long as it satisfies the constraints $$p_1(x) = p_0'(x)$$ and $$[p_0 (f^* g' - (f^*)' g)]_a^b = 0$$. - -Let us ignore the latter constraint for now (it will return later), -and focus on the former: what if $$\hat{L}$$ does not satisfy $$p_0' \neq p_1$$? -We multiply it by an unknown $$p(x) \neq 0$$, and divide by $$p_0(x) \neq 0$$: +We now demand that the derivative $$p'(x)$$ of the unknown $$p(x)$$ satisfies: $$\begin{aligned} - \frac{p(x)}{p_0(x)} \hat{L} \{u\} = p(x) u'' + p(x) \frac{p_1(x)}{p_0(x)} u' + p(x) \frac{p_2(x)}{p_0(x)} u + p'(x) + = p(x) \frac{p_1(x)}{p_2(x)} + \quad \implies \quad + \frac{p_1(x)}{p_2(x)} \dd{x} + = \frac{1}{p(x)} \dd{p} \end{aligned}$$ -We now define $$q(x)$$, -and demand that the derivative $$p'(x)$$ of the unknown $$p(x)$$ satisfies: +Taking the indefinite integral of this differential equation +yields an expression for $$p(x)$$: $$\begin{aligned} - q(x) = p(x) \frac{p_2(x)}{p_0(x)} - \qquad - p'(x) = p(x) \frac{p_1(x)}{p_0(x)} + \int \frac{p_1(x)}{p_2(x)} \dd{x} + = \int \frac{1}{p} \dd{p} + = \ln\!\big( p(x) \big) + \quad \implies \quad + \boxed{ + p(x) + = \exp\!\bigg( \int \frac{p_1(x)}{p_2(x)} \dd{x} \bigg) + } \end{aligned}$$ -The latter is a differential equation for $$p(x)$$, which we solve by integration: +We define an additional function $$q(x)$$ +based on the last term of $$(p / p_2) \hat{L}$$ shown above: $$\begin{aligned} - \frac{p_1(x)}{p_0(x)} \dd{x} - &= \frac{1}{p(x)} \dd{p} - \\ - \implies \quad - \int \frac{p_1(x)}{p_0(x)} \dd{x} - &= \int \frac{1}{p} \dd{p} - = \ln\!\big( p(x) \big) - \\ - \implies \qquad\qquad - p(x) - &= \exp\!\bigg( \int \frac{p_1(x)}{p_0(x)} \dd{x} \bigg) + \boxed{ + q(x) + \equiv p(x) \frac{p_0(x)}{p_2(x)} + } + = \frac{p_0(x)}{p_2(x)} \exp\!\bigg( \int \frac{p_1(x)}{p_2(x)} \dd{x} \bigg) \end{aligned}$$ -Now that we have $$p(x)$$ and $$q(x)$$, we can define a new operator $$\hat{L}_p$$ as follows: +When rewritten using $$p$$ and $$q$$, +the modified operator $$(p / p_2) \hat{L}$$ looks like this: $$\begin{aligned} - \hat{L}_p \{u\} - = \frac{p}{p_0} \hat{L} \{u\} + \frac{p}{p_2} \hat{L} \{u\} = p u'' + p' u' + q u = (p u')' + q u \end{aligned}$$ This is the self-adjoint form from earlier! -So even if $$p_0' \neq p_1$$, any second-order linear operator with $$p_0(x) \neq 0$$ -can easily be put in self-adjoint form. - -This general form is known as the **Sturm-Liouville operator** $$\hat{L}_{SL}$$, -where $$p(x)$$ and $$q(x)$$ are nonzero real functions of the variable $$x \in [a,b]$$: +So even if $$p_1 \neq p_2'$$, any second-order linear operator +with $$p_2(x) \neq 0$$ can easily be made self-adjoint. +The resulting general form is called the **Sturm-Liouville operator** $$\hat{L}_\mathrm{SL}$$, +for nonzero $$p(x)$$: $$\begin{aligned} \boxed{ - \hat{L}_{SL} \{u(x)\} - = \frac{d}{dx}\Big( p(x) \frac{du}{dx} \Big) + q(x) u(x) - = \hat{L}_{SL}^\dagger \{u(x)\} + \begin{aligned} + \hat{L}_\mathrm{SL} \{u(x)\} + &= \hat{L}_\mathrm{SL}^\dagger \{u(x)\} + \\ + &= \Big( p(x) \: u'(x) \Big)' + q(x) \: u(x) + \end{aligned} } \end{aligned}$$ +Still subject to the constraint $$\big[ p (f^* g' - (f^*)' g) \big]_a^b = 0$$ +such that $$\inprod{f}{\hat{L}_\mathrm{SL} g} = \inprod{\hat{L}_\mathrm{SL}^\dagger f}{g}$$. + ## Eigenvalue problem -A **Sturm-Liouville problem** (SLP) is analogous to a matrix eigenvalue problem, -where $$w(x)$$ is a real weight function, $$\lambda$$ is the **eigenvalue**, -and $$u(x)$$ is the corresponding **eigenfunction**: +An eigenvalue problem of $$\hat{L}_\mathrm{SL}$$ +is called a **Sturm-Liouville problem** (SLP). +The goal is to find the **eigenvalues** $$\lambda$$ +and corresponding **eigenfunctions** $$u(x)$$ that fulfill: $$\begin{aligned} \boxed{ - \hat{L}_{SL}\{u(x)\} = - \lambda w(x) u(x) + \hat{L}_\mathrm{SL}\{u(x)\} = - \lambda \: w(x) \: u(x) } \end{aligned}$$ -Necessarily, $$w(x) > 0$$ except in isolated points, where $$w(x) = 0$$ is allowed; -the point is that any inner product $$\inprod{f}{w g}$$ may never be zero due to $$w$$'s fault. -Furthermore, the convention is that $$u(x)$$ cannot be trivially zero. +Where $$w(x)$$ is a real weight function satisfying $$w(x) > 0$$ for $$x \in \,\,]a, b[$$. +By convention, the trivial solution $$u = 0$$ is not valid. +Some authors have the opposite sign for $$\lambda$$ and/or $$w$$. -In our derivation of $$\hat{L}_{SL}$$, -we removed a boundary term to get self-adjointness. -Consequently, to have a valid SLP, the boundary conditions for -$$u(x)$$ must be as follows, otherwise the operator cannot be self-adjoint: +In our derivation of $$\hat{L}_\mathrm{SL}$$ above, +we imposed the constraint $$\big[ p (f^* g' - (f')^* g) \big]_a^b = 0$$ to ensure that +$$\inprod{\hat{L}_\mathrm{SL}^\dagger f}{g} = \inprod{f}{\hat{L}_\mathrm{SL} g}$$. +Consequently, to have a valid SLP, +the boundary conditions (BCs) on $$u$$ must be such that, +for any two (possibly identical) eigenfunctions $$u_m$$ and $$u_n$$, we have: $$\begin{aligned} - \Big[ p(x) \big( u^*(x) u'(x) - (u'(x))^* u(x) \big) \Big]_a^b = 0 + \Big[ p(x) \big( u_m^*(x) \: u_n'(x) - \big(u_m'(x)\big)^* u_n(x) \big) \Big]_a^b = 0 \end{aligned}$$ -There are many boundary conditions (BCs) which satisfy this requirement. -Some notable ones are listed here non-exhaustively: +There are many boundary conditions that satisfy this requirement. +Some notable ones are listed non-exhaustively below. +Verify for yourself that these work: + **Dirichlet BCs**: $$u(a) = u(b) = 0$$ + **Neumann BCs**: $$u'(a) = u'(b) = 0$$ @@ -190,108 +201,103 @@ Some notable ones are listed here non-exhaustively: + **Periodic BCs**: $$p(a) = p(b)$$, $$u(a) = u(b)$$, and $$u'(a) = u'(b)$$ + **Legendre "BCs"**: $$p(a) = p(b) = 0$$ -Once this requirement is satisfied, Sturm-Liouville theory gives us -some very useful information about $$\lambda$$ and $$u(x)$$. -From the definition of an SLP, we know that, given two arbitrary (and possibly identical) -eigenfunctions $$u_n$$ and $$u_m$$, the following must be satisfied: - -$$\begin{aligned} - 0 = \hat{L}_{SL}\{u_n\} + \lambda_n w u_n = \hat{L}_{SL}\{u_m^*\} + \lambda_m^* w u_m^* -\end{aligned}$$ - -We subtract these expressions, multiply by the eigenfunctions, and integrate: +If this is fulfilled, Sturm-Liouville theory gives us +useful information about $$\lambda$$ and $$u$$. +By definition, the following must be satisfied +for two arbitrary eigenfunctions $$u_m$$ and $$u_n$$: $$\begin{aligned} 0 - &= \int_a^b u_m^* \big(\hat{L}_{SL}\{u_n\} + \lambda_n w u_n\big) - u_n \big(\hat{L}_{SL}\{u_m^*\} + \lambda_m^* w u_m^*\big) \:dx + &= \hat{L}_\mathrm{SL}\{u_m^*\} + \lambda_m^* w u_m^* \\ - &= \int_a^b u_m^* \hat{L}_{SL}\{u_n\} - u_n \hat{L}_{SL}\{u_m^*\} + u_n u_m^* w (\lambda_n - \lambda_m^*) \:dx + &= \hat{L}_\mathrm{SL}\{u_n\} + \lambda_n w u_n \end{aligned}$$ -Rearranging this a bit reveals that these are in fact three inner products: +We multiply each by the other eigenfunction, +subtract the results, and integrate: $$\begin{aligned} - \int_a^b u_m^* \hat{L}_{SL}\{u_n\} - u_n \hat{L}_{SL}\{u_m^*\} \:dx - &= (\lambda_m^* - \lambda_n) \int_a^b u_n u_m^* w \:dx + 0 + &= \int_a^b u_m^* \big(\hat{L}_\mathrm{SL}\{u_n\} + \lambda_n w u_n\big) + - u_n \big(\hat{L}_\mathrm{SL}\{u_m^*\} + \lambda_m^* w u_m^*\big) \dd{x} \\ - \inprod{u_m}{\hat{L}_{SL} u_n} - \inprod{\hat{L}_{SL} u_m}{u_n} - &= (\lambda_m^* - \lambda_n) \inprod{u_m}{w u_n} + &= \int_a^b u_m^* \hat{L}_\mathrm{SL}\{u_n\} - u_n \hat{L}_\mathrm{SL}\{u_m^*\} + + (\lambda_n - \lambda_m^*) u_m^* w u_n \dd{x} + \\ + &= \inprod{u_m}{\hat{L}_\mathrm{SL} u_n} - \inprod{\hat{L}_\mathrm{SL} u_m}{u_n} + + (\lambda_n - \lambda_m^*) \inprod{u_m}{w u_n} \end{aligned}$$ -The operator $$\hat{L}_{SL}$$ is self-adjoint by definition, -so the left-hand side vanishes, leaving us with: +The operator $$\hat{L}_\mathrm{SL}$$ is self-adjoint of course, +so the first two terms vanish, leaving us with: $$\begin{aligned} 0 - &= (\lambda_m^* - \lambda_n) \inprod{u_m}{w u_n} + &= (\lambda_n - \lambda_m^*) \inprod{u_m}{w u_n} \end{aligned}$$ -When $$m = n$$, the inner product $$\inprod{u_n}{w u_n}$$ is real and positive -(assuming $$u_n$$ is not trivially zero, in which case it would be disqualified anyway). -In this case we thus know that $$\lambda_n^* = \lambda_n$$, -i.e. the eigenvalue $$\lambda_n$$ is real for any $$n$$. - -When $$m \neq n$$, then $$\lambda_m^* - \lambda_n$$ may or may not be zero, -depending on the degeneracy. If there is no degeneracy, we -see that $$\inprod{u_m}{w u_n} = 0$$, i.e. the eigenfunctions are orthogonal. +When $$m = n$$, we get $$\inprod{u_n}{w u_n} > 0$$, +so the equation is only satisfied if $$\lambda_n^* = \lambda_n$$, +meaning the eigenvalue $$\lambda_n$$ is real for any $$n$$. +When $$m \neq n$$, then $$\lambda_n - \lambda_m^*$$ +may or may not be zero depending on the degeneracy. +If there is no degeneracy, then $$\lambda_n - \lambda_m^* \neq 0$$, +meaning $$\inprod{u_m}{w u_n} = 0$$, i.e. the eigenfunctions are orthogonal. +In case of degeneracy, manual orthogonalization is needed, +which is guaranteed to be doable using the [Gram-Schmidt method](/know/concept/gram-schmidt-method/). -In case of degeneracy, manual orthogonalization is needed, but as it turns out, -this is guaranteed to be doable, using e.g. the [Gram-Schmidt method](/know/concept/gram-schmidt-method/). - -In conclusion, **a Sturm-Liouville problem has real eigenvalues $$\lambda$$, -and all the corresponding eigenfunctions $$u(x)$$ are mutually orthogonal**: +In conclusion, an SLP has **real eigenvalues** +and **orthogonal eigenfunctions**: for all $$m$$, $$n$$: $$\begin{aligned} \boxed{ - \inprod{u_m(x)}{w(x) u_n(x)} - = \inprod{u_n}{w u_n} \delta_{nm} + \lambda_n \in \mathbb{R} + } + \qquad\qquad + \boxed{ + \inprod{u_m}{w u_n} = A_n \delta_{nm} } \end{aligned}$$ -When you're solving a differential eigenvalue problem, -knowing that all eigenvalues are real is a *huge* simplification, +When solving a differential eigenvalue problem, +knowing that all eigenvalues are real is a huge simplification, so it is always worth checking whether you are dealing with an SLP. -Another useful fact of SLPs is that they always -have an infinite number of discrete eigenvalues. -Furthermore, the eigenvalues always ascend to $$+\infty$$; -in other words, there always exists a *lowest* eigenvalue $$\lambda_0 > -\infty$$, -known as the **ground state**. +Another useful fact: +it turns out that SLPs always have an infinite number of *discrete* eigenvalues. +Furthermore, there always exists a *lowest* eigenvalue $$\lambda_0 > -\infty$$, +called the **ground state**. -## Completeness +## Complete basis -Not only are the eigenfunctions $$u_n(x)$$ of an SLP orthogonal, they -also form a **complete basis**, meaning that any well-behaved function $$f(x)$$ can be -expanded as a **generalized Fourier series** with coefficients $$a_n$$: +Not only are an SLP's eigenfunctions orthogonal, +they also form a **complete basis**, meaning any well-behaved $$f(x)$$ +can be expanded as a **generalized Fourier series** with coefficients $$a_n$$: $$\begin{aligned} \boxed{ f(x) = \sum_{n = 0}^\infty a_n u_n(x) - \quad \mathrm{for}\: x \in ]a, b[ + \quad \mathrm{for} \: x \in \,\,]a, b[ } \end{aligned}$$ -This series will converge si |