A Force Acting on a Wave Packet Leads to the Equation of Motion.
We calculate the motion of an electron in an electric field using only the Schrödinger equation and compare the center of its wave packet with the solution of the equation of motion.
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波の計算の結果は、運動方程式の答えとどれだけ違うか
The equations explain the phenomenon
1We obtain the wave relation in the absence of a force.ħω=ħ²k²2m……
3We find the speed of the wave packet.v=dωdk=ħkm……
4We express the speed in terms of momentum.v=pm……
5We introduce a force.iħ ∂ψ∂t=−(ħ²2m) ∂²ψ∂x²− F x ψ……
6The force term rotates the wave's phase at a rate that depends on position.ψ→ψ × eiFxt/ħ……
7The wavenumber increases with time.k(t)=k₀+Ftħ……
8Force changes momentum.dpdt=F……
9The equation of motion follows.mdvdt=F……
The following section considers forces that vary with position.
10We derive the equation for the average when the force varies with position.d⟨p⟩dt=⟨F(x)⟩，d⟨x⟩dt=⟨p⟩m……
11For a narrow wave packet, the average force approaches the force at its center.m d²⟨x⟩dt²≒F(⟨x⟩)+½ F″(⟨x⟩) σ²……
12For a spring, the forces agree regardless of the packet width.F=−kx⇒F″=0……
13At a narrow barrier, the motion departs from the equation of motion.σ≳Distance over which the force varies⇒The wave packet splits into 2 parts.……
14Wave packets of massive objects spread negligibly.Spreading time≒2mσ²ħ……
What this chapter has shown
The force rotates the phase at a rate that depends on position, reducing the spacing between wave crests and giving ħ dk/dt = F. Combined with v = ħk/m, this shows that the wave packet center obeys m dv/dt = F.
This holds when the wave packet is narrow compared with the distance over which the force varies. At a narrow barrier, the packet splits into 2 parts.
In everyday lifeIn an electron microscope, electric and magnetic fields deflect an electron beam to form an image. Because the centers of the electron wave packets obey the equation of motion, the microscope can be designed by calculating electron trajectories.
Detailed summary
The force term in the Schrödinger equation shifts the wave's phase by a different amount at each position, changing the wavenumber according to ħ dk/dt = F. Since the wave packet's velocity is v = ħk/m, its center obeys m dv/dt = F. When the force varies with position, d⟨p⟩/dt = ⟨F(x)⟩ holds exactly. If the wave packet is narrow compared with the length scale over which the force varies, this relation reduces to the equation of motion.