1 of 20

## What's hot

Kronig penny model_computational_phyics
Kronig penny model_computational_phyicsNeerajKumarMeena5

spherical coordinates system
spherical coordinates systemPankaj Nakum

Quantum Theory of Magnetism- Heisenberg Model
Quantum Theory of Magnetism- Heisenberg ModelSara Khorshidian

Introduction to perturbation theory, part-1
Introduction to perturbation theory, part-1Kiran Padhy

Fermi dirac basic part 1

Langevin theory of Paramagnetism
Langevin theory of ParamagnetismDr. HAROON

Schrodinger equation and its applications: Chapter 2
Schrodinger equation and its applications: Chapter 2Dr.Pankaj Khirade

Lecture06h Frequency Dependent Transport5.ppt

Bi-linear transformation 2
Bi-linear transformation 2Buvaneswari

Time Independent Perturbation Theory, 1st order correction, 2nd order correction
Time Independent Perturbation Theory, 1st order correction, 2nd order correctionJames Salveo Olarve

Rectangular waveguides
Rectangular waveguideskhan yaseen

The wkb approximation
The wkb approximationZahid Mehmood

Finite difference method
Finite difference methodDivyansh Verma

Maxwell's equation
Maxwell's equationAL- AMIN

Second order homogeneous linear differential equations
Second order homogeneous linear differential equations Viraj Patel

### What's hot(20)

6 slides
6 slides

Poisson's equation 2nd 4
Poisson's equation 2nd 4

Kronig penny model_computational_phyics
Kronig penny model_computational_phyics

spherical coordinates system
spherical coordinates system

Quantum Theory of Magnetism- Heisenberg Model
Quantum Theory of Magnetism- Heisenberg Model

Introduction to perturbation theory, part-1
Introduction to perturbation theory, part-1

Fourier series and transforms
Fourier series and transforms

Fermi dirac basic part 1
Fermi dirac basic part 1

Langevin theory of Paramagnetism
Langevin theory of Paramagnetism

Legendre functions
Legendre functions

L9 fuzzy implications
L9 fuzzy implications

Schrodinger equation and its applications: Chapter 2
Schrodinger equation and its applications: Chapter 2

Lecture06h Frequency Dependent Transport5.ppt
Lecture06h Frequency Dependent Transport5.ppt

Bi-linear transformation 2
Bi-linear transformation 2

Time Independent Perturbation Theory, 1st order correction, 2nd order correction
Time Independent Perturbation Theory, 1st order correction, 2nd order correction

Rectangular waveguides
Rectangular waveguides

The wkb approximation
The wkb approximation

Finite difference method
Finite difference method

Maxwell's equation
Maxwell's equation

Second order homogeneous linear differential equations
Second order homogeneous linear differential equations

## Similar to Helmholtz equation (Motivations and Solutions)

Uniformity of the Local Convergence of Chord Method for Generalized Equations
Uniformity of the Local Convergence of Chord Method for Generalized EquationsIOSR Journals

Magnetic Monopoles, Duality and SUSY.pptx
Magnetic Monopoles, Duality and SUSY.pptxHassaan Saleem

Outgoing ingoingkleingordon ghp
Outgoing ingoingkleingordon ghpfoxtrot jp R

Outgoing ingoingkleingordon
Outgoing ingoingkleingordonfoxtrot jp R

Outgoing ingoingkleingordon julups
Outgoing ingoingkleingordon julupsfoxtrot jp R

Lesson 3: Problem Set 4
Lesson 3: Problem Set 4Kevin Johnson

Cauchy-Euler Equation.pptx
Cauchy-Euler Equation.pptxSalarBasheer

Gauge Theory for Beginners.pptx
Gauge Theory for Beginners.pptxHassaan Saleem

Outgoing ingoingkleingordon 8th_jun19sqrd
Outgoing ingoingkleingordon 8th_jun19sqrdfoxtrot jp R

Conformal Boundary conditions
Conformal Boundary conditionsHassaan Saleem

Lecture 1.6 further graphs and transformations of quadratic equations
Lecture 1.6 further graphs and transformations of quadratic equationsnarayana dash

Lane_emden_equation_solved_by_HPM_final

Review of Seiberg Witten duality.pptx
Review of Seiberg Witten duality.pptxHassaan Saleem

Outgoing ingoingkleingordon spvmforminit_proceedfrom12dec18
Outgoing ingoingkleingordon spvmforminit_proceedfrom12dec18foxtrot jp R

Outgoing ingoingkleingordon spvmforminit_proceedfrom
Outgoing ingoingkleingordon spvmforminit_proceedfromfoxtrot jp R

### Similar to Helmholtz equation (Motivations and Solutions)(20)

Uniformity of the Local Convergence of Chord Method for Generalized Equations
Uniformity of the Local Convergence of Chord Method for Generalized Equations

Magnetic Monopoles, Duality and SUSY.pptx
Magnetic Monopoles, Duality and SUSY.pptx

Outgoing ingoingkleingordon ghp
Outgoing ingoingkleingordon ghp

Outgoing ingoingkleingordon
Outgoing ingoingkleingordon

Outgoing ingoingkleingordon julups
Outgoing ingoingkleingordon julups

Lesson 3: Problem Set 4
Lesson 3: Problem Set 4

Cauchy-Euler Equation.pptx
Cauchy-Euler Equation.pptx

Waveguides
Waveguides

odes1.pptx
odes1.pptx

Gauge Theory for Beginners.pptx
Gauge Theory for Beginners.pptx

C0560913
C0560913

Outgoing ingoingkleingordon 8th_jun19sqrd
Outgoing ingoingkleingordon 8th_jun19sqrd

Conformal Boundary conditions
Conformal Boundary conditions

Lecture 1.6 further graphs and transformations of quadratic equations
Lecture 1.6 further graphs and transformations of quadratic equations

Lane_emden_equation_solved_by_HPM_final
Lane_emden_equation_solved_by_HPM_final

Review of Seiberg Witten duality.pptx
Review of Seiberg Witten duality.pptx

Outgoing ingoingkleingordon spvmforminit_proceedfrom12dec18
Outgoing ingoingkleingordon spvmforminit_proceedfrom12dec18

Outgoing ingoingkleingordon spvmforminit_proceedfrom
Outgoing ingoingkleingordon spvmforminit_proceedfrom

Triple Integral
Triple Integral

《Queensland毕业文凭-昆士兰大学毕业证成绩单》
《Queensland毕业文凭-昆士兰大学毕业证成绩单》rnrncn29

Pests of Bengal gram_Identification_Dr.UPR.pdf
Pests of Bengal gram_Identification_Dr.UPR.pdfPirithiRaju

Pests of soyabean_Binomics_IdentificationDr.UPR.pdf
Pests of soyabean_Binomics_IdentificationDr.UPR.pdfPirithiRaju

well logging & petrophysical analysis.pptx
well logging & petrophysical analysis.pptxzaydmeerab121

Pests of Blackgram, greengram, cowpea_Dr.UPR.pdf
Pests of Blackgram, greengram, cowpea_Dr.UPR.pdfPirithiRaju

Pests of castor_Binomics_Identification_Dr.UPR.pdf
Pests of castor_Binomics_Identification_Dr.UPR.pdfPirithiRaju

GLYCOSIDES Classification Of GLYCOSIDES Chemical Tests Glycosides
GLYCOSIDES Classification Of GLYCOSIDES Chemical Tests GlycosidesNandakishor Bhaurao Deshmukh

User Guide: Capricorn FLX™ Weather Station
User Guide: Capricorn FLX™ Weather StationColumbia Weather Systems

Servosystem Theory / Cybernetic Theory by Petrovic
Servosystem Theory / Cybernetic Theory by PetrovicAditi Jain

CHROMATOGRAPHY PALLAVI RAWAT.pptx
CHROMATOGRAPHY PALLAVI RAWAT.pptxpallavirawat456

The Sensory Organs, Anatomy and Function
The Sensory Organs, Anatomy and FunctionJadeNovelo1

Pests of jatropha_Bionomics_identification_Dr.UPR.pdf
Pests of jatropha_Bionomics_identification_Dr.UPR.pdfPirithiRaju

Thermodynamics ,types of system,formulae ,gibbs free energy .pptx
Thermodynamics ,types of system,formulae ,gibbs free energy .pptxuniversity

User Guide: Pulsar™ Weather Station (Columbia Weather Systems)
User Guide: Pulsar™ Weather Station (Columbia Weather Systems)Columbia Weather Systems

LESSON PLAN IN SCIENCE GRADE 4 WEEK 1 DAY 2
LESSON PLAN IN SCIENCE GRADE 4 WEEK 1 DAY 2AuEnriquezLontok

User Guide: Orion™ Weather Station (Columbia Weather Systems)
User Guide: Orion™ Weather Station (Columbia Weather Systems)Columbia Weather Systems

Charateristics of the Angara-A5 spacecraft launched from the Vostochny Cosmod...
Charateristics of the Angara-A5 spacecraft launched from the Vostochny Cosmod...Christina Parmionova

Interferons.pptx.
Interferons.pptx.

《Queensland毕业文凭-昆士兰大学毕业证成绩单》
《Queensland毕业文凭-昆士兰大学毕业证成绩单》

Pests of Bengal gram_Identification_Dr.UPR.pdf
Pests of Bengal gram_Identification_Dr.UPR.pdf

Pests of soyabean_Binomics_IdentificationDr.UPR.pdf
Pests of soyabean_Binomics_IdentificationDr.UPR.pdf

AZOTOBACTER AS BIOFERILIZER.PPTX
AZOTOBACTER AS BIOFERILIZER.PPTX

well logging & petrophysical analysis.pptx
well logging & petrophysical analysis.pptx

Pests of Blackgram, greengram, cowpea_Dr.UPR.pdf
Pests of Blackgram, greengram, cowpea_Dr.UPR.pdf

Pests of castor_Binomics_Identification_Dr.UPR.pdf
Pests of castor_Binomics_Identification_Dr.UPR.pdf

GLYCOSIDES Classification Of GLYCOSIDES Chemical Tests Glycosides
GLYCOSIDES Classification Of GLYCOSIDES Chemical Tests Glycosides

User Guide: Capricorn FLX™ Weather Station
User Guide: Capricorn FLX™ Weather Station

Servosystem Theory / Cybernetic Theory by Petrovic
Servosystem Theory / Cybernetic Theory by Petrovic

CHROMATOGRAPHY PALLAVI RAWAT.pptx
CHROMATOGRAPHY PALLAVI RAWAT.pptx

The Sensory Organs, Anatomy and Function
The Sensory Organs, Anatomy and Function

Pests of jatropha_Bionomics_identification_Dr.UPR.pdf
Pests of jatropha_Bionomics_identification_Dr.UPR.pdf

Thermodynamics ,types of system,formulae ,gibbs free energy .pptx
Thermodynamics ,types of system,formulae ,gibbs free energy .pptx

User Guide: Pulsar™ Weather Station (Columbia Weather Systems)
User Guide: Pulsar™ Weather Station (Columbia Weather Systems)

LESSON PLAN IN SCIENCE GRADE 4 WEEK 1 DAY 2
LESSON PLAN IN SCIENCE GRADE 4 WEEK 1 DAY 2

User Guide: Orion™ Weather Station (Columbia Weather Systems)
User Guide: Orion™ Weather Station (Columbia Weather Systems)

Charateristics of the Angara-A5 spacecraft launched from the Vostochny Cosmod...
Charateristics of the Angara-A5 spacecraft launched from the Vostochny Cosmod...

### Helmholtz equation (Motivations and Solutions)

• 1. Helmholtz equation (Motivation and solutions) Muhammad Hassaan Saleem (PHYMATHS)
• 2. Helmholtz equation The Helmholtz equation is given as; 𝛻2 𝜓 + 𝑘2 𝜓 = 0 𝑘 ∈ ℛ Where 𝜓(𝑥, 𝑦, 𝑧) is a function of 𝑥, 𝑦 and 𝑧 𝑘2 is a constant Note : 𝜓 should have continuous first and second partial derivatives with respect to 𝑥, 𝑦, 𝑧
• 4. Motivation for Helmholtz equation 1) Schrodinger equation for free particle The Schrodinger equation for a free particle of mass 𝑚 and energy 𝐸 is given as; − ℏ2 2𝑚 𝛻2 𝜓 𝑥, , 𝑦, 𝑧 = 𝐸𝜓(𝑥, 𝑦, 𝑧) Where 𝜓(𝑥, 𝑦, 𝑧) is the particle wavefunction It can be written as; 𝛻2 𝜓 𝑥, 𝑦, 𝑧 + 2𝑚𝐸 ℏ2 𝜓 𝑥, 𝑦, 𝑧 = 0 This is the Helmholtz equation for (if 𝐸 ≥ 0) if we set 2𝑚𝐸 ℏ2 = 𝑘2 , 𝑘 ∈ ℛ
• 5. Motivation for Helmholtz equation 2) Waves with eikonal time dependence The wave equation is given as; 𝛻2 𝑢 𝑥, 𝑦, 𝑧, 𝑡 − 1 𝑣2 𝜕2 𝑢 𝑥, 𝑦, 𝑧, 𝑡 𝜕𝑡2 = 0 If time dependence of u(𝑥, 𝑦, 𝑧, 𝑡) is given by the eikonal factor i.e. 𝑒 𝑖𝜔𝑡, then u x, y, z, t = e𝑖𝜔𝑡 𝜓 𝑥, 𝑦, 𝑧 and the wave equation can be written as; 𝛻2 𝜓 𝑥, 𝑦, 𝑧 + 𝜔 𝑣 2 𝜓 𝑥, 𝑦, 𝑧 = 0 which is Helmholtz equation for 𝑘2 = 𝜔 𝑣 2 . This 𝑘 is known as the wavenumber in the context of wave equation.
• 7. Solution in Cartesian coordinates The explicit form of Helmholtz equation in Cartesian coordinates is given as; 𝜕2 𝜓 𝑥, 𝑦, 𝑧 𝜕𝑥2 + 𝜕2 𝜓 𝑥, 𝑦, 𝑧 𝜕𝑦2 + 𝜕2 𝜓 𝑥, 𝑦, 𝑧 𝜕𝑧2 + 𝑘2 𝜓 𝑥, 𝑦, 𝑧 = 0 (𝑖) We now employ a method known as method of separation of variables. We assume a solution of the form; 𝜓 𝑥, 𝑦, 𝑧 = 𝑋 𝑥 𝑌 𝑦 𝑍 𝑧 (𝑖𝑖) Where 𝑋(𝑥) is a function of 𝑥 only 𝑌(𝑦) is a function of 𝑦 only 𝑍(𝑧) is a function of 𝑧 only
• 8. Solution in Cartesian coordinates Now we use (𝑖𝑖) in (𝑖). This changes partial derivatives to total derivatives and then, we divide whole equation by 𝑋 𝑥 𝑌 𝑦 𝑍(𝑧). We then get, 1 𝑋 𝑑2 𝑋 𝑑𝑥2 + 1 𝑌 𝑑2 𝑌 𝑑𝑦2 + 1 𝑍 𝑑2 𝑍 𝑑𝑧2 + 𝑘2 = 0 We can rearrange it as 1 𝑋 𝑑2 𝑋 𝑑𝑥2 + 1 𝑌 𝑑2 𝑌 𝑑𝑦2 + 𝑘2 = − 1 𝑍 𝑑2 𝑍 𝑑𝑧2 Now, LHS is independent of 𝑧 while RHS is dependant only on 𝑧. So, this equation can satisfied only if both sides are equal to a constant.
• 9. Solution in Cartesian coordinates This means that 1 𝑍 𝑑2 𝑍 𝑑𝑧2 = − 𝑛 2 ⇒ 𝑍 𝑧 = 𝐴 𝑧 sin 𝑛𝑧 + 𝐵𝑧 cos(𝑛𝑧) where 𝑛2 is a positive constant while 𝐴 𝑧 and 𝐵𝑧 are arbitrary constants. Note : We have choosen 𝑛2 to be positive because it gives a solution which doesn’t grow exponentially and thus, gives physically more relevant solution (e.g. for the case when 𝜓 represents a wavefunction). We can do the same procedure for 𝑋(𝑥) and 𝑌(𝑦) functions.
• 10. Solution in Cartesian coordinates When we perform the same procedure for 𝑋 𝑥 , Y(y) and 𝑍(𝑧) functions, we get; 𝑋 𝑥 = 𝑋𝑙(𝑥) = 𝐴 𝑥 sin 𝑙𝑥 + 𝐵𝑥 cos(𝑙𝑥) 𝑌 𝑦 = 𝑌 𝑚 𝑦 = 𝐴 𝑦 sin 𝑚𝑦 + 𝐵𝑦 cos(𝑚𝑦) 𝑍 𝑧 = 𝑍 𝑛 𝑧 = 𝐴 𝑧 sin 𝑛𝑧 + 𝐵𝑧 cos(𝑛𝑧) Where 𝑙, 𝑚, 𝑛, 𝐴 𝑥,𝑦,𝑧 , 𝐵 𝑥,𝑦,𝑧 are constants. We see that each solution are labelled by a parameter i.e. 𝑙, 𝑚 and 𝑛. With these equations, we also get the equation 𝑙2 + 𝑚2 + 𝑛2 = 𝑘2 We can thus deduce that 𝜓𝑙𝑚𝑛 𝑥, 𝑦, 𝑧 = 𝑋𝑙 𝑥 𝑌 𝑚 𝑦 𝑍 𝑛(𝑧) We can see that 𝜓 is labelled now by three parameters.
• 11. General solution The constants 𝐴 𝑥,𝑦,𝑧 and 𝐵 𝑥,𝑦,𝑧 are determined by boundary conditions of the differential equation. Because of the relation 𝑙2 + 𝑚2 + 𝑛2 = 𝑘2 We know that only two parameters among 𝑙, 𝑚 and 𝑛 are independent. We choose them to be 𝑙 and 𝑚. So, we can write the general solution to the equation as 𝜓 𝑥, 𝑦, 𝑧 = 𝑙,𝑚 𝑎𝑙𝑚 𝜓𝑙𝑚𝑛(𝑥, 𝑦, 𝑧) where 𝑎𝑙𝑚 are constants. They too, are choosen so that the boundary conditions are satisfied. This usually leads to discrete values of 𝑙, 𝑚. We will see an example in the next slide.
• 12. An example: Particle in a cubic box. In problem of particle in a cubic box of length 𝑎, we have 𝑘2 = 2𝑚𝐸/ℏ2 . The Schrodinger equation was quoted as a Helmholtz equation already and this problem has the following boundary conditions 𝜓 0, 𝑦, 𝑧 = 𝜓 𝑎, 𝑦, 𝑧 = 0 𝜓 𝑥, 0, 𝑧 = 𝜓 𝑥, 𝑎, 𝑧 = 0 𝜓 𝑥, 𝑦, 0 = 𝜓 𝑥, 𝑦, 𝑎 = 0 Because of the vanishing of the 𝜓 function on 𝑥 = 0, 𝑦 = 0 and 𝑧 = 0, we deduce that 𝐵𝑥 = 0, 𝐵𝑦 = 0, 𝐵𝑧 = 0 and thus, we write the solution as 𝜓 𝑥, 𝑦, 𝑧 = 𝑙,𝑚 𝑎𝑙𝑚 sin 𝑙𝑥 sin 𝑚𝑦 sin(𝑛𝑧) Where we have absorbed the arbitrary constants 𝐴 𝑥, 𝐴 𝑦 and 𝐴 𝑧 into 𝑎𝑙𝑚.
• 13. An example : Particle in a cubic box Applying the remaining boundary conditions, we get; sin 𝑙𝑎 = 0 ⇒ 𝑙 = 𝑐1 𝜋 𝑎 , sin 𝑚𝑎 = 0 ⇒ 𝑚 = 𝑐2 𝜋 𝑎 , sin 𝑛𝑎 = 0 ⇒ 𝑛 = 𝑐3 𝜋 𝑎 Where 𝑐 1,2,3 = 1,2,3,4,5, … Due to the relation 𝑙2 + 𝑚2 + 𝑛2 = 𝑘2 , we have 2𝑚𝐸 ℏ2 = 𝜋2 𝑎2 𝑐1 2 + 𝑐2 2 + 𝑐3 2 ⇒ 𝐸 = ℏ2 𝜋2 2𝑚𝑎2 𝑐1 2 + 𝑐2 2 + 𝑐3 2 This gives the energy levels which the particle can attain.
• 14. Solution in cylindrical coordinates (a sketch) We consider the Helmholtz equation in cylindrical coordinates 𝑟, 𝜃, 𝑧 for the function 𝜓 𝑟, 𝜃, 𝑧 . The 𝛻2 operator in these coordinates is given as 𝛻2 = 𝜕2 𝜕𝑟2 + 1 𝑟 𝜕 𝜕𝑟 + 1 𝑟2 𝜕2 𝜕𝜃2 + 𝜕2 𝜕𝑧2 We can do the separation 𝜓 𝑟, 𝜃, 𝑧 = 𝑅 𝑟 Θ 𝜃 Z(z). Using the above expression for the 𝛻2 operator and the method of separation of variables we can derive the solution of the equation.
• 15. Solution in cylindrical coordinates (a sketch) After some simplification, we can get the following equations 𝑑2 𝑍 𝑑𝑧2 = 𝑙2 𝑍, 𝑑2Θ 𝑑𝜃2 = −𝑚2 𝜃, 𝑟 𝑑 𝑑𝑟 𝑟 𝑑𝑃 𝑑𝑟 + 𝑛2 𝑟2 − 𝑚2 𝑃 = 0 Several comments are in order  In the first equation, 𝑙2 is choosen to have an exponentially decaying solution.  In the second equation −𝑚2 is choosen to have a periodic solution  The third equation is the Bessel equation with argument 𝑛𝑟. Along these equations, we also get 𝑛2 = 𝑙2 + 𝑘2 so, there are again two independent parameters among 𝑙, 𝑚 and 𝑛. Here too, boundary conditions are required to specify the particular solution of the equation.
• 16. Solution in cylindrical coordinates (a sketch) A sidenote The Helmholtz equation can again be solved in cylindrical coordinates by using the method of separation of variables if we replace 𝑘2 as 𝑘2 → f r + g 𝜃 r2 + h(z) Where f 𝑟 , 𝑔(𝜃) and ℎ(𝑧) are arbitrary differentiable functions of 𝑟, 𝜃 and 𝑧.
• 17. Solution in spherical coordinates (a sketch) We can use the expression for 𝛻2 in spherical coordinates (𝑟, 𝜃, 𝜙) i.e. 𝛻2 = 𝜕2 𝜕𝑟2 + 2 𝑟 𝜕 𝜕𝑟 + 1 𝑟2 𝜕2 𝜕𝜃2 + cos 𝜃 r2 sin 𝜃 𝜕 𝜕𝜃 + 1 𝑟2 sin2 𝜃 𝜕2 𝜕𝜙2 With it, we can make the separation 𝜓 𝑟, 𝜃, 𝜙 = 𝑅 𝑟 Θ 𝜃 Φ(𝜙) and use the method of separation of variables to get the equations for 𝑅, Θ and Φ.
• 18. Solution in spherical coordinates (a sketch) We get these equations; 𝚽 equation 𝑑2 Φ 𝑑𝜙2 = −𝑚2 Φ(𝜙) The constant −𝑚2 is choosen to make Φ(𝜙) a periodic function of 𝜙. 𝚯 equation sin2 𝜃 𝑑2 Θ 𝑑 cos 𝜃 2 − 2 cos 𝜃 𝑑Θ 𝑑 cos 𝜃 + 𝑙 𝑙 + 1 − 𝑚2 sin2 𝜃 Θ = 0 This is an associated Legendre equation in the argument cos 𝜃. The term 𝑙(𝑙 + 1) (where 𝑙 is an integer) comes from the fact that this equation has non singular solutions only if we have a term 𝑙(𝑙 + 1) there.
• 19. Solution in spherical coordinates (a sketch) R equation The 𝑅 equation is 𝑑2 𝑅 𝑑𝑟2 + 2 𝑟 𝑑𝑅 𝑑𝑟 + 𝑘2 𝑅 − 𝑙 𝑙 + 1 𝑅 𝑟2 = 0 This is the spherical Bessel equation with the argument 𝑘𝑟. So, we can use the known solutions of all these equations to write the solutions in spherical coordinates. Sidenote: The Helmholtz equation can still be solved by separation of variables if we replace 𝑘2 by 𝑘2 → 𝑓 𝑟 + 𝑔 𝜃 𝑟2 + ℎ 𝜙 𝑟2 sin2 𝜃 Where 𝑓 𝑟 , 𝑔(𝜃) and ℎ(𝜙) are arbitrary functions of 𝑟, 𝜃 and 𝜙.
Current LanguageEnglish
Español
Portugues
Français
Deutsche