Phy5645/Particle bouncing on the floor in Earth's Gravitational field
We need to solve the Schrödinger equation for this problem,
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\frac{\hbar^2}{2m}\frac{d^2\psi(z)}{dz^2}+mgz\psi(z)=E\psi(z),}
subject to the boundary condtions, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi(0)=0} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi(z\to\infty)=0.}
If we make the change of variable, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=\left (\frac{2}{mg^2\hbar^2}\right )^\frac{1}{3}(mgz-E),} then we can reduce the above equation to
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{d^2\phi(x)}{dx^2}-x\phi(x)=0.}
This is a standard differential equation, known as the Airy equation. The only physical solution to this equation is
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi(x)=B\text{Ai}(x),\!}
where
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \text{Ai}(x)=\frac{1}{\pi}\int_{0}^{\infty}\cos\left (\frac{t^3}{3}+xt\right )\,dt.}
At Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z=0,\,x=-\left (\frac{2}{mg^2\hbar^2}\right )^\frac{1}{3}E.}
The boundary condition Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi(0)=0} yields Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi\left [-\left (\frac{2}{mg^2\hbar^2}\right )^\frac{1}{3}E\right ]=0,}
or
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \text{Ai}\left [-\left (\frac{2}{mg^2\hbar^2}\right )^\frac{1}{3}E\right ]=0.}
The Airy function has zeros only at certain values Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R_n} such that
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \text{Ai}(R_n)=0\!} for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n=0,1,2,3\ldots}
The first few roots of the Airy function are:
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R_0=-2.338,\,R_1=-4.088,\,R_2=-5.521,\,R_3=-6.787,\,\ldots}
The boundary condition Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi(0)=0} therefore results in a discrete set of energy levels that can be expressed in terms of the roots of the Airy function; i.e.,
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \text{Ai}\left [-\left (\frac{2}{mg^2\hbar^2}\right )^\frac{1}{3}E\right ]=0\; \Rightarrow \; -\left (\frac{2}{mg^2\hbar^2}\right )^\frac{1}{3}E_n=R_n.}
Thus the discrete energy levels of the particle are given by
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_n=-\left (\frac{mg^2\hbar^2}{2}\right )^\frac{1}{3}R_n,}
and the associated wave functions are Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi_n(z)=B_n\text{Ai}\left [-\left (\frac{2m^2g^2}{\hbar^2}\right )^\frac{1}{3}z-R_n\right ].}
The first few energy levels of the particle are:
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_0=2.338\left (\frac{mg^2\hbar^2}{2}\right )^\frac{1}{3},\,E_1=4.088\left (\frac{mg^2\hbar^2}{2}\right )^\frac{1}{3},\,E_2=5.521\left (\frac{mg^2\hbar^2}{2}\right )^\frac{1}{3},\,E_3=6.787\left (\frac{mg^2\hbar^2}{2}\right )^\frac{1}{3}}
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