- #1

Lengalicious

- 163

- 0

## Homework Statement

See attachment.

## Homework Equations

## The Attempt at a Solution

So I have shown that the plane wave sol'n satisfies the Klein-Gordon equation by subbing in and reducing the equation to:

[tex]E^2 = p^2c^2 + m^2c^4[/tex]

which reduces to:

[tex]E = pc[/tex]

for an m = 0 particle, however I don't really know how to show that this would correspond with the wave traveling at the speed of light.

Next I am asked to act on the wave function with the Hamiltonian and momentum operators, I assume I should act with the relativistic Hamiltonian so:

[tex]\hat{H} = (-\hbar^2\frac{\partial^2 }{\partial x^2}c^2 + m^2c^4)^{1/2}[/tex]

So I get:

[tex]\hat{H}\psi = \sqrt{4\hbar^2c^2k^2 + m^2c^4}(e^{i(wt-kx)})[/tex]

So:

[tex]E = \sqrt{4\hbar^2c^2k^2 + m^2c^4}[/tex]

[tex]h\nu = \sqrt{4\hbar^2c^2k^2 + m^2c^4}[/tex]

Reduces to:

[tex]-3h^2\nu^2 = m^2c^4[/tex]

So I'm not sure what I have done wrong,

now for the momentum:

[tex]\hat{p}\psi = -i\hbar\frac{\partial }{\partial x}(e^{i(wt-kx)})[/tex]

Reduces to:

[tex]= -\hbar k(e^{i(wt-kx)})[/tex]

[tex]\frac{h}{\lambda} = -\hbar k[/tex]

[tex]\frac{h}{\lambda} = -\frac{h}{\lambda}[/tex]

Once again something has gone wrong, not sure what. . .

On the very last bit 2.(a) I basically don't know how to show that it does not satisfy the energy-mass-momentum relation. I have tried separating components and solving for E

_{3}with,

[tex](m_ec,\bar{0}) = ((|\bar{p}_2|^2 + m_e^2c^2)^{1/2},\bar{p}_2) + (E_3,\bar{p}_3)[/tex]

to no avail, so I'm pretty lost on this one any general advice or direction would help.

Thanks in advance!