Overview
The de Broglie hypothesis was that matter that was normally understood to be particles are actually waves with a wavelength ({% \lambda %}) given by
{% \lambda = \frac{h}{p} %}
where
- {% h %} is Planks constant
- {% p %} is the momentum
Schrodinger Wave Function and Momentum
Postulating that a particle with a definite momentum is made from a wavefunction with a wavelength, one can postulate that one such wave function could look like
{% f_k(x) = e^{ikx} %}
Here, we relabel the wavelength to be the traditional notation, {% k %}. Then, if we apply the derivative operator
to such a wavefunction we get
{% \frac{d}{dx} f_k(x) = ik f_k(x) %}
That is, if we take momentum operator to be
{% \hat{P} = -i \frac{d}{dx} %}
we see that the wavelength (momentum) is an eigenvalue of this operator.
see larkoski pg 17
Generator of Translations
Taylor Expansion
{% f(x+a) = f(x) + a \frac{df}{dx} + \frac{a^2}{2} \frac{d^2f}{dx^2} + ... %}
{% f(x+a) \approx (1 + \frac{a}{N} \frac{d}{dx})^N f(x) = e^{a\frac{d}{dx}} f(x) %}