>[!summary] If we assume the particle in a box only exist in standing waves from [[Quantization of Atomic Orbital's]] > Quantized momentum: $p_n = \frac{nh}{2L}$ > Quantized Energy: $E = \frac{h^2}{8L^2m}$ For [[Particle in a Box (1D)]] we allow only certain standing waves $\lambda = \frac{2L}{n}$, $p = \frac{h}{\lambda}$ , $E = \frac{p^2}{2m}$ For ground state: ![[Pasted image 20250327105229.png]] $\lambda = \frac{2l}{1}$ So we can find the momentum and energy $p = \frac{h}{2L}$ The momentum refers to the momentum a particle must have when measured to from this standing wave. ## General form for momentum for any nth value >[!info] Assumptions This is found from deriving the nth value from [[Quantization of Atomic Orbital's]] Where n = 1,2,3,.. $p_n = \frac{nh}{2L}$ When we dont measure it has a quantum wave having both +p and -p but when we measure it collapes into one standing wave $\begin{array}{c} E = \frac{p_n^2}{2m} \\ E = \frac{h^2}{8L^2m} \\ \end{array}$ Like momentum this is the energy a particle would have to have if we measured it with this wavelength