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Key Equations

Normalization condition in one dimension P(x=,+)=|Ψ(x,t)|2dx=1P(x=,+)=|Ψ(x,t)|2dx=1
Probability of finding a particle in a narrow interval of position in one dimension (x,x+dx)(x,x+dx) P(x,x+dx)=Ψ*(x,t)Ψ(x,t)dxP(x,x+dx)=Ψ*(x,t)Ψ(x,t)dx
Expectation value of position in one dimension x=Ψ*(x,t)xΨ(x,t)dxx=Ψ*(x,t)xΨ(x,t)dx
Heisenberg’s position-momentum uncertainty principle ΔxΔp2ΔxΔp2
Heisenberg’s energy-time uncertainty principle ΔEΔt2ΔEΔt2
Schrӧdinger’s time-dependent equation 22m2Ψ(x,t)x2+U(x,t)Ψ(x,t)=iΨ(x,t)t22m2Ψ(x,t)x2+U(x,t)Ψ(x,t)=iΨ(x,t)t
General form of the wave function for a time-independent potential in one dimension Ψ(x,t)=ψ(x)eiωtΨ(x,t)=ψ(x)eiωt
Schrӧdinger’s time-independent equation 22md2ψ(x)dx2+U(x)ψ(x)=Eψ(x)22md2ψ(x)dx2+U(x)ψ(x)=Eψ(x)
Schrӧdinger’s equation (free particle) 22m2ψ(x)x2=Eψ(x)22m2ψ(x)x2=Eψ(x)
Allowed energies (particle in box of length L) En=n2π222mL2,n=1,2,3,...En=n2π222mL2,n=1,2,3,...
Stationary states (particle in a box of length L) ψn(x)=2LsinnπxL,n=1,2,3,...ψn(x)=2LsinnπxL,n=1,2,3,...
Potential-energy function of a harmonic oscillator U(x)=12mω2x2U(x)=12mω2x2
Schrӧdinger equation (harmonic oscillator) 22md2ψ(x)dx2+12mω2x2ψ(x)=Eψ(x)22md2ψ(x)dx2+12mω2x2ψ(x)=Eψ(x)
The energy spectrum En=(n+12)ω,n=0,1,2,3,...En=(n+12)ω,n=0,1,2,3,...
The energy wave functions ψn(x)=Nneβ2x2/2Hn(βx),n=0,1,2,3,...ψn(x)=Nneβ2x2/2Hn(βx),n=0,1,2,3,...
Potential barrier U(x)={0,whenx<0U0,when0xL0,whenx>LU(x)={0,whenx<0U0,when0xL0,whenx>L
Definition of the transmission coefficient T(L,E)=|ψtra(x)|2|ψin(x)|2T(L,E)=|ψtra(x)|2|ψin(x)|2
A parameter in the transmission coefficient β2=2m2(U0E)β2=2m2(U0E)
Transmission coefficient, exact T(L,E)=1cosh2βL+(γ/2)2sinh2βLT(L,E)=1cosh2βL+(γ/2)2sinh2βL
Transmission coefficient, approximate T(L,E)=16EU0(1EU0)e2βLT(L,E)=16EU0(1EU0)e2βL
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