- #1
davijcanton
- 3
- 0
Hi,
I'm trying to resolve a problem (17-2) of Pauling's book (Introduction to Quantum Mechanics ), but I'm not achieving this integration. So, I ask for your help. The problem says:
Calculate [itex]\overline{p_{z}²}[/itex] (where p[itex]_{z}[/itex] = momentum in z direction and [itex]\overline{x}[/itex] = average value of x for example) for a harmonic oscillator (in cylindrical coordinates) in a state ( n',m,n[itex]_{z}[/itex] ) represented by:
where:
N = const
[itex]\alpha[/itex] = cost = 4[itex]\pi[/itex]²*mass/h² * (freq[itex]_{0}[/itex])
[itex]\alpha[/itex]z = cost = 4[itex]\pi[/itex]²*mass/h * (freq[itex]_{z}[/itex])
H[itex]_{nz}[/itex] = zn Hermite polynomial
F(k) = finite polynomial in k[itex]^{|m|}[/itex] I think this polynomial don't have larger influence in the result.
Thenks in advance.
See you soon,
I'm trying to resolve a problem (17-2) of Pauling's book (Introduction to Quantum Mechanics ), but I'm not achieving this integration. So, I ask for your help. The problem says:
Calculate [itex]\overline{p_{z}²}[/itex] (where p[itex]_{z}[/itex] = momentum in z direction and [itex]\overline{x}[/itex] = average value of x for example) for a harmonic oscillator (in cylindrical coordinates) in a state ( n',m,n[itex]_{z}[/itex] ) represented by:
[itex]\Psi_{n',m,n_{z}}[/itex]([itex]\rho[/itex],[itex]\varphi[/itex],z) = N e[itex]^{im\varphi}[/itex] e[itex]^{-\alpha\rho²/2 }[/itex] F[itex]_{|m|,n'}[/itex]([itex]\sqrt{\alpha}[/itex] [itex]\rho[/itex]) e[itex]^{-\alpha_{z}z²/2}[/itex] H[itex]_{nz}[/itex] ([itex]\sqrt{\alpha_{z}}[/itex] z)
where:
N = const
[itex]\alpha[/itex] = cost = 4[itex]\pi[/itex]²*mass/h² * (freq[itex]_{0}[/itex])
[itex]\alpha[/itex]z = cost = 4[itex]\pi[/itex]²*mass/h * (freq[itex]_{z}[/itex])
H[itex]_{nz}[/itex] = zn Hermite polynomial
F(k) = finite polynomial in k[itex]^{|m|}[/itex] I think this polynomial don't have larger influence in the result.
Thenks in advance.
See you soon,
Davis