History of Particle in a box

	Revision 5 . . (edit) November 28, 2001 5:01 am by (logged).254.0.xxx [* ^^]
	Revision 4 . . November 22, 2001 6:05 am by AxelBoldt [Hamilton operator contains h_bar squared.]
	Revision 3 . . (edit) October 9, 2001 2:52 pm by The Cunctator

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Difference (from prior major revision) (minor diff, author diff)

Changed: 13c13

:-h_bar/2m d2ψ/dx2 + V(x)ψ = Eψ    (1)

:-h2/2m d2ψ/dx2 + V(x)ψ = Eψ    (1)

Changed: 15c15

where h_bar is Plancks constant divided by 2π, m is the mass of the particle, ψ is the wavefunction that we want to find, V(x) is a function describing the potential at each point x and E is the energy. For the case of the particle in a 1-dimensional box of length L, the potential is zero inside the box, but rises abruptly to infinity at x = 0 and x = L. Thus for the region inside the box V(x) = 0 and Equation 1 reduces to:

where h is Plancks constant divided by 2π, m is the mass of the particle, ψ is the wavefunction that we want to find, V(x) is a function describing the potential at each point x and E is the energy. For the case of the particle in a 1-dimensional box of length L, the potential is zero inside the box, but rises abruptly to infinity at x = 0 and x = L. Thus for the region inside the box V(x) = 0 and Equation 1 reduces to:

Changed: 17c17

:-h_bar/2m d2ψ/dx2 = Eψ    (2)

:-h2/2m d2ψ/dx2 = Eψ    (2)

Changed: 22c22

:E = k2 h_bar2 / 2m    (3)

:E = k2 h2 / 2m    (3)

Changed: 48c48

:En = n2h_bar2π2/2mL2 =n2h2/8mL2     n = 1,2,...    (9)

:En = n2h2π2/2mL2 =n2h2/8mL2     n = 1,2,...    (9)

Changed: 61c61

:-h_bar/2m (d2ψ/dx2 + d2ψ/dy2) = Eψ    (11)

:-h2/2m (d2ψ/dx2 + d2ψ/dy2) = Eψ    (11)

Changed: 71c71

:-h_bar/2m (Yd2X/dx2 + Xd2Y/dy2) = EXY    (13)

:-h2/2m (Yd2X/dx2 + Xd2Y/dy2) = EXY    (13)

Changed: 75c75

:-h_bar/2m (X"/X + Y"/Y) = E     (14)

:-h2/2m (X"/X + Y"/Y) = E     (14)

Changed: 79c79

:-h_bar/2m X"/X = Ex and -h_bar/2m Y"/Y = Ey      (15)

:-h2/2m X"/X = Ex and -h2/2m Y"/Y = Ey      (15)

Changed: 83c83

:-h_bar/2m d2X/dx2 = ExX     (16)

:-h2/2m d2X/dx2 = ExX     (16)

Changed: 85c85

:-h_bar/2m d2Y/dy2 = EyY     (17)

:-h2/2m d2Y/dy2 = EyY     (17)

Changed: 103c103

:Enx,ny = {(nx/Lx)2 + (ny/Ly)2 + (nz/Lz)2} h2/8m    (23)

:Enx,ny,nz = {(nx/Lx)2 + (ny/Ly)2 + (nz/Lz)2} h2/8m    (23)