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Okay, so I know that the wavelength of an electron is 5e-7m and I am asked to calculate its minimum velocity and hence minimum energy.

Calculating the minimum velocity via $\lambda = h/p$ gives 1454.67 m/s. Next step is to calculate the associated energy.

The actual answer given to me is $E_{min} = 1/2 mv^2 = 9.71e-25J$, whereas I decided to calculate it using:

$E_{min} = hf = hv/\lambda_{max} = 1.93e-24J$.

Why does my answer differ?

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For a De Broglie wave, in the equation $E = hf$, $E$ is the total relativistic energy, considering rest mass and kinetic energy ($E^2 = p^2c^2 + m^2c^4$). You cannot compare this to the kinetic energy alone.

Also, for a De Broglie wave, the phase velocity and group velocity are different. The particle velocity corresponds to the group velocity, whereas in your final equation $v$ represents the phase velocity.

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