Helmholtz equation (Motivations and Solutions)

Solutions of Helmholtz equation in cartesian, cylindrical and spherical coordinates are discussed and the applications to the problem of a quantum mechanical particle in a cubical box is discussed.

1. Helmholtz equation (Motivation and solutions) Muhammad Hassaan Saleem (PHYMATHS)

2. Helmholtz equation The Helmholtz equation is given as; Where is a function of and is a constant Note : should have continuous first and second partial derivatives with respect to  

3. Motivations for Helmholtz equation

4. Motivation for Helmholtz equation Schrodinger equation for free particle The Schrodinger equation for a free particle of mass and energy is given as; Where is the particle wavefunction It can be written as; This is the Helmholtz equation for (if ) if we set  

5. Motivation for Helmholtz equation 2) Waves with eikonal time dependence The wave equation is given as; If time dependence of is given by the eikonal factor i.e. , then and the wave equation can be written as; which is Helmholtz equation for . This is known as the wavenumber in the context of wave equation.  

6. Solutions of Helmholtz equation

7. Solution in Cartesian coordinates The explicit form of Helmholtz equation in Cartesian coordinates is given as; We now employ a method known as method of separation of variables. We assume a solution of the form; Where is a function of only is a function of only is a function of only  

8. Solution in Cartesian coordinates Now we use in . This changes partial derivatives to total derivatives and then, we divide whole equation by . We then get, We can rearrange it as Now, LHS is independent of while RHS is dependant only on . So, this equation can satisfied only if both sides are equal to a constant.  

9. Solution in Cartesian coordinates This means that where is a positive constant while and are arbitrary constants. Note : We have choosen 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 represents a wavefunction). We can do the same procedure for and functions.  

10. Solution in Cartesian coordinates When we perform the same procedure for and functions, we get; Where are constants. We see that each solution are labelled by a parameter i.e. and . With these equations, we also get the equation We can thus deduce that We can see that is labelled now by three parameters.  

11. General solution The constants and are determined by boundary conditions of the differential equation. Because of the relation We know that only two parameters among and are independent. We choose them to be and . So, we can write the general solution to the equation as where are constants. They too, are choosen so that the boundary conditions are satisfied. This usually leads to discrete values of . 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 , we have . The Schrodinger equation was quoted as a Helmholtz equation already and this problem has the following boundary conditions Because of the vanishing of the function on and we deduce that and thus, we write the solution as Where we have absorbed the arbitrary constants and into .  

13. An example : Particle in a cubic box Applying the remaining boundary conditions, we get; , Where Due to the relation , we have 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 for the function The operator in these coordinates is given as We can do the separation . Using the above expression for the 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 Several comments are in order In the first equation, is choosen to have an exponentially decaying solution. In the second equation is choosen to have a periodic solution The third equation is the Bessel equation with argument . Along these equations, we also get so, there are again two independent parameters among and . 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 as Where and are arbitrary differentiable functions of and .  

17. Solution in spherical coordinates (a sketch) We can use the expression for in spherical coordinates i.e. With it, we can make the separation and use the method of separation of variables to get the equations for and .  

18. Solution in spherical coordinates (a sketch) We get these equations; equation The constant is choosen to make a periodic function of . equation This is an associated Legendre equation in the argument . The term (where is an integer) comes from the fact that this equation has non singular solutions only if we have a term there.  

19. Solution in spherical coordinates (a sketch) R equation The equation is This is the spherical Bessel equation with the argument . 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 by Where and are arbitrary functions of and .  

20. Thank you for listening Questions?