Suppose we have a particle with mass m and energy E in a gravitational field V(z)=−mgz. How can I find the wave function ψ(z)?
It should have an integral form on dp. Any help would be appreciated.
What I've tried
One way to solve the problem is use of change of variable
x := (ℏ22m2g)2/32mℏ2(mgz−E)
we can reduce Schroedinger equation to
d2ϕdx2−xϕ(x) = 0
This is a standard equation, its solution is given by ϕ(x) = B Ai(x)
where Ai is the Airy function. But my solution should be (not exactly) like this:
ψ(z)=N∫∞−∞dpexp[(Emg+z)p−p36m2g]
Answer
[p22m+V(iℏddp)]ϕ(p)=Eϕ(p)
[p22m+(−mg)(iℏddp)]ϕ(p)=Eϕ(p)
1iℏmg(p22m−E)ϕ(p)=ϕ(p)dp
When integrate we have: iℏmg(Ep−p36m)=Lnϕ(p)ϕ(po)
ϕ(p)=ϕ(p0)eEmgp−p36m2g
ψ(z)=∫dpeipz/ℏϕ(p)
ψ(z)=ϕ(p0)∫∞−∞dpei/ℏ[(Emg+z)p−p36m2g]
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