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Hello,

I wondered if someone could help me show / give hints on how to show the (simplified) elastic differential equation (below) is related to the classical Schrodinger equation (in quantum mechanics)? I am a maths undergraduate.

(simplified) elastic differential equation: [tex]$d^2y/dx^2 + \lambda^2g(x)y=0$[/tex]. Assumtion: the value of Lambda tends to infinity. Also: [tex]$g(x)>=g(0)>0$[/tex]

What i believe is the classical schrodinger equation (?):

[tex]$[[-h^2/2m][pd^2/pdx^2] + V(x)$](\phi)=E(\phi)[/tex],

I have got so far: [tex]$[[-h^2/2m][pd^2/pdx^2] + [1/2][kx^2]](\phi)=E(\phi)$[/tex]

Assuming (perhaps incorrectly that potential function V(x) is: [tex]$V(x)=(1/2)kx^2$[/tex] for system, [tex]$\phi$[/tex] is the wavefunction, and [tex]$[[-h^2/2m][pd^2/pdx^2] + V(x)[/tex] is Hamiltonian operator.

I don't know how to show they are similar. - Main concern is in schrod. equation (PDE) phi is being operated on, wheras in elastic eqn (ODE) y(x) is either a part of a product not operated on or is in other term. I can see that k = lambda^2.

Thank you

dev00790

## Homework Statement

I wondered if someone could help me show / give hints on how to show the (simplified) elastic differential equation (below) is related to the classical Schrodinger equation (in quantum mechanics)? I am a maths undergraduate.

**2. Homework Equations**- (i had some trouble with tex/latex while writing these. By "pd" I mean partial derivative(simplified) elastic differential equation: [tex]$d^2y/dx^2 + \lambda^2g(x)y=0$[/tex]. Assumtion: the value of Lambda tends to infinity. Also: [tex]$g(x)>=g(0)>0$[/tex]

What i believe is the classical schrodinger equation (?):

[tex]$[[-h^2/2m][pd^2/pdx^2] + V(x)$](\phi)=E(\phi)[/tex],

**3. My Attempt**I have got so far: [tex]$[[-h^2/2m][pd^2/pdx^2] + [1/2][kx^2]](\phi)=E(\phi)$[/tex]

Assuming (perhaps incorrectly that potential function V(x) is: [tex]$V(x)=(1/2)kx^2$[/tex] for system, [tex]$\phi$[/tex] is the wavefunction, and [tex]$[[-h^2/2m][pd^2/pdx^2] + V(x)[/tex] is Hamiltonian operator.

I don't know how to show they are similar. - Main concern is in schrod. equation (PDE) phi is being operated on, wheras in elastic eqn (ODE) y(x) is either a part of a product not operated on or is in other term. I can see that k = lambda^2.

Thank you

dev00790

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