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A massless particle situated in a 1D infinite square well with momentum only in the direction of quantum confinement (the x direction):

[tex]E_t \psi (x) = \hbar \omega \psi (x) = i \hbar \frac{\partial}{\partial t} \psi (x)[/tex]

[tex]p(x) \psi(x) = \hbar k_x \psi (x) = -i \hbar \frac{\partial}{\partial x} \psi (x)[/tex]

Integration by substitution:

[tex]E_k(x) \psi(x) = c p(x) \psi(x) = -i \hbar c \frac{\partial}{\partial x} \psi (x)[/tex]

[tex]E_t \psi (x) = E_k(x) \psi(x) + V_u(x) \psi (x)[/tex]

Integration by substitution:

[tex]i \hbar \frac{\partial}{\partial t} \psi (x) = -i \hbar c \frac{\partial}{\partial x} \psi (x) + V_u(x) \psi (x)[/tex]

Massless Schrödinger equation:

[tex]\boxed{i \hbar \frac{\partial}{\partial t} \psi (x) = -i \hbar c \frac{\partial}{\partial x} \psi (x) + V_u(x) \psi (x)}[/tex]

Mass Schrödinger equation:

[tex]i \hbar \frac{\partial}{\partial t} \psi (x) = -\frac{\hbar^2}{2m} \frac{\partial^2 \psi(x)}{\partial x^2} + V(x) \psi(x)[/tex]

Is my solution for the massless Schrödinger equation correct?

Reference:

http://en.wikipedia.org/wiki/Schr%C3%B6dinger_equation" [Broken]

http://en.wikipedia.org/wiki/Particle_in_a_box" [Broken]

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# Massless Schrödinger equation

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