University Physics Volume 3

# Key Equations

### 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 $〈x〉=∫−∞∞Ψ*(x,t)xΨ(x,t)dx〈x〉=∫−∞∞Ψ*(x,t)xΨ(x,t)dx$ Heisenberg’s position-momentum uncertainty principle $ΔxΔp≥ℏ2ΔxΔp≥ℏ2$ Heisenberg’s energy-time uncertainty principle $ΔEΔt≥ℏ2ΔEΔt≥ℏ2$ Schrӧdinger’s time-dependent equation $−ℏ22m∂2Ψ(x,t)∂x2+U(x,t)Ψ(x,t)=iℏ∂Ψ(x,t)∂t−ℏ22m∂2Ψ(x,t)∂x2+U(x,t)Ψ(x,t)=iℏ∂Ψ(x,t)∂t$ General form of the wave function for a time-independent potential in one dimension $Ψ(x,t)=ψ(x)e−iωtΨ(x,t)=ψ(x)e−iωt$ Schrӧdinger’s time-independent equation $−ℏ22md2ψ(x)dx2+U(x)ψ(x)=Eψ(x)−ℏ22md2ψ(x)dx2+U(x)ψ(x)=Eψ(x)$ Schrӧdinger’s equation (free particle) $−ℏ22m∂2ψ(x)∂x2=Eψ(x)−ℏ22m∂2ψ(x)∂x2=Eψ(x)$ Allowed energies (particle in box of length L) $En=n2π2ℏ22mL2,n=1,2,3,...En=n2π2ℏ22mL2,n=1,2,3,...$ 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) $−ℏ22md2ψ(x)dx2+12mω2x2ψ(x)=Eψ(x)−ℏ22md2ψ(x)dx2+12mω2x2ψ(x)=Eψ(x)$ The energy spectrum $En=(n+12)ℏω,n=0,1,2,3,...En=(n+12)ℏω,n=0,1,2,3,...$ The energy wave functions $ψn(x)=Nne−β2x2/2Hn(βx),n=0,1,2,3,...ψn(x)=Nne−β2x2/2Hn(βx),n=0,1,2,3,...$ Potential barrier $U(x)={0,whenx<0U0,when0≤x≤L0,whenx>LU(x)={0,whenx<0U0,when0≤x≤L0,whenx>L$ 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 $β2=2mℏ2(U0−E)β2=2mℏ2(U0−E)$ Transmission coefficient, exact $T(L,E)=1cosh2βL+(γ/2)2sinh2βLT(L,E)=1cosh2βL+(γ/2)2sinh2βL$ Transmission coefficient, approximate $T(L,E)=16EU0(1−EU0)e−2βLT(L,E)=16EU0(1−EU0)e−2βL$
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