# Quantum Animals

A penguin exhibiting quantum behaviour

It is known that animals cannot be explained using classical physics, since they tunnel through walls. The chickens thus exhibit quantum behavior and should be described with quantum mechanics.

Time-independent Schrödinger equation

${\displaystyle {\hat {H}}\vert \Psi \rangle =E\vert \Psi \rangle }$

The Schrödinger Equation gives us the behavior of wave function of the animal, which describes probabilistically how the animal will behave. According to the Born interpretation, ${\displaystyle \langle \Psi (x,t)|\Psi (x,t)\rangle }$ describes the probability that the chicken will be at ${\displaystyle x}$, but the exact outcome of the experiment cannot be known.

## Estimation of mass of chicken

We know that chickens tunnel through 1m thick walls, so their De Broglie Wavelength should be on the order of 1m, and their speed is on the order of ${\displaystyle 0.1{\frac {\text{m}}{\text{s}}}}$. We know that

${\displaystyle \lambda ={\frac {h}{p}}={\frac {h}{mv}}\iff m={\frac {h}{v\lambda }}}$

Plugging our values in and with ${\displaystyle h=6.626\ 070\ 15\times 10^{-34}\ {\text{J}}{\cdot }{\text{s}}}$, we get

${\displaystyle m={\frac {h}{0.1{\frac {{\text{m}}^{2}}{\text{s}}}}}=6.626\ 070\ 15\times 10^{-33}\ {\text{kg}}}$

This shows us that the chickens are very very light.

## Behaviour of chicken in pen

This problem requires that we know the Hamiltonian operator ${\displaystyle {\hat {H}}=V(r)-{\frac {\hbar }{2m}}\nabla ^{2}}$ of our chicken pen, where ${\displaystyle V(r)}$ is the potential of the pen, so we can solve the Schrödinger equation for this problem. We can assume that the potential is piecewise constant, zero inside the pen and ${\displaystyle V_{0}>0}$ outside (finite potential well model), so ${\displaystyle V(r)={\begin{cases}0&|r|\leq L\\V_{0}&|r|>L\end{cases}}.}$

### Inside the pen

Inside the pen, we have ${\displaystyle V(r)=0}$. Plugging this into the Schrödinger equation gives us

${\displaystyle -{\frac {\hbar }{2m}}\nabla ^{2}\Psi =E\Psi }$

Define ${\displaystyle k:={\frac {\sqrt {2mE}}{\hbar }}}$. Since our problem is one-dimensional, we can solve this second-order differential equation with a function

${\displaystyle \Psi (x)=A\sin(kx)+B\cos(kx)}$

for some constants ${\displaystyle A,B\in \mathbb {C} }$

### Outside the pen

Outside the pen, we have ${\displaystyle V(r)>0}$. The calculation of the Hamiltonian and the solving of the Schrödinger equation is left as an exercise to the reader.

This is an excerpt of the upcoming book "Quantum Mechanics of Animals"

Book Cover