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  • Deriving Electron’s Kinetic Energy from Uncertainty Principle

    Say, electron is confined within a region of size ‘r’.

    The uncertainty principle tells us:

    ΔxΔp≥ℏ2\Delta x \Delta p \geq \frac{\hbar}{2}

    using Δx∼r\Delta x \sim r

    Δp≥ℏ2r\Delta p \geq \frac{\hbar}{2r}

    If the electron is confined to a region of size r, its momentum can not be smaller than its uncertainty in its momentum

    ∴p∼Δp\therefore p \sim \Delta p

    or,

    p≥ℏ2rp \geq \frac{\hbar}{2r}

    Using an order-of-magnitude approximation

    p∼ℏrp \sim \frac{\hbar}{r}

    substituting this estimate into classical expression for kinetic energy we get,

    KE=p22m∼ℏ22mr2KE=\frac{p^2}{2m} \sim \frac{\hbar^2}{2mr^2}

    KE=ℏ22mr2KE=\frac{\hbar^2}{2mr^2}