Which of the following two formulas is the correct wavelength expression of a relativistic electron?

Question

$$ \lambda=\dfrac{h}{m\gamma v } \tag{01}\label{01} $$ or $$ \lambda=\dfrac{hv}{m\gamma c^2 } \tag{02}\label{02} $$

Edit: The energy of an electron is, by special relativity $$ E=\gamma mc^2 \tag{03}\label{03} $$ where $$ \gamma = \left(1-\frac{v^2}{c^2}\right)^{-1/2} \tag{04}\label{04} $$ The frequency $f$ satisfies the Planck relation: $$ E = hf \tag{05}\label{05} $$ and the frequency and wavelength $\:\lambda\:$ of any wave must satisfy $$ f\lambda = v \tag{06}\label{06} $$ Put all of those together and I get $$ \lambda = \dfrac{v}{f} = \dfrac{hv}{E} = \dfrac{hv}{\gamma mc^2} \tag{07}\label{07} $$

Answer 1

Reference : My answer there About de Broglie relations, what exactly is E? Its energy of what?.

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The correct wavelength expression of a relativistic electron is (01), while (02) is wrong. This is due to the wrong equation (06). The wavelength $\:\lambda\:$ and the frequency $\:f\:$ are characteristics of the "superluminal" plane phase wave associated to the "subluminal" particle. The speed of this phase wave is $\:c^2/\upsilon$, so the correct equation (06) is \begin{equation} f\lambda\boldsymbol{=}\dfrac{c^2}{\upsilon} \tag{A-01}\label{A-01} \end{equation} Hence the correct equation (07) is \begin{equation} \lambda \boldsymbol{=}\dfrac{c^2}{f\upsilon}\boldsymbol{=}\dfrac{c^2}{\left(E/h\right)\upsilon}\boldsymbol{=}\dfrac{c^2}{\left(\gamma m c^2/h\right)\upsilon}\boldsymbol{=}\dfrac{h}{\gamma m \upsilon} \tag{A-02}\label{A-02} \end{equation} that is equation (01) of the question.