# Time dependence of de Broglie wavelength of an electron with initial velocity $V\hat{\imath}$ and in a uniform magnetic field $B\hat{\jmath}$

I am supposed to comment on the time dependence of de Broglie wavelength of an electron with initial velocity $$V\hat{\imath}$$ and in a uniform magnetic field $$B\hat{\jmath}$$.

Obviously since the electron is a negatively charged particle force on it will be in $$-\hat{k}$$-direction.

And the velocity in $$z$$-direction can be written as, $$\vec{V}_z=\frac{1}{2}\frac{eVB}{m}t^{2}(-\hat{k})$$

There will be no velocity in $$y$$-direction as there is no acceleration and the velocity in $$x$$-direction will remain constant at $$V\hat{\imath}$$

So quite clearly the velocity in $$-z$$-direction will increase as time increases and that will increase the momentum and since $$\lambda\propto\frac{1}{\mathrm{momentum}}$$, the de Broglie wavelength would decrease with time. But the book says that the de Broglie wavelength would remain constant with time. The reason it provides is: "Force is perpendicular to both $$\vec{V}$$ and $$\vec{B}$$ and the magnitude of $$\vec{V}$$ will not change. It means momentum will not change."

$$|\mathbf{v}|$$ does not change, as $$\frac{d}{dt} \mathbf{v}^2=2 \, \mathbf{v} \cdot\dot{\mathbf{v}}=\frac{2q}{m} \mathbf{v} \cdot (\mathbf{v} \times \mathbf{B})=0$$.