The vibrations of atoms in a molecule can be modeled as a quantum mechanical spring. Even in the lowest-energy state, the position is not confined to a single point.
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基底状態の位置の幅を半分にした直後、平均エネルギーは何倍か？
基底状態の位置の幅を2倍にした直後、平均エネルギーは何倍か？
The equations explain the phenomenon
1The spring potential energy is substituted.−(ħ²2m) φ″+½ m ω² x² φ=E φ……
2A Gaussian function is substituted to find the energy.φ₀ ∝ e−mωx²/2ħ，E₀=½ ħω……
3The excited-state energies are examined.En=ħω (n + ½)……
4The order of the position and momentum operators is reversed.x p − p x=iħ……
5The relation between the position and momentum spreads is derived.Δx · Δp≧ħ2……
6A lower bound on the energy is found for states with mean position and momentum 0.⟨E⟩≧ħ²8mΔx²+½ m ω² Δx²……
7The spread that minimizes the energy bound F(Δx) is found.Minimum of F(Δx)=½ ħω，Δx=σ₀=√(ħ2mω)……
8The energy immediately after changing the position spread is calculated.⟨E⟩=2 E₀+E₀8=2.125 E₀……
Further topics: Motion of expectation values and the hydrogen atom
9The time evolution of the mean position and momentum is derived.d⟨x⟩dt=⟨p⟩m，d⟨p⟩dt=−⟨dVdx⟩……
The following section treats 3-dimensional, spherically symmetric states with angular momentum 0. This requires more than replacing V in the 1-dimensional equation.
10The hydrogen atom's energy levels are examined.En=−13.6 eVn²……
What this chapter has shown
When the mean position and momentum are 0, narrowing the position spread increases the lower bound on the kinetic energy, while widening it increases the potential energy. Using Δx·Δp ≧ ħ/2 to find a lower bound on their sum gives a minimum of ½ħω. The ground-state probability distribution does not change with time.
2.125 times⟨E⟩÷E₀ immediately after the operation
Halving the position spread and doubling the momentum spread of the ground state increases the mean energy by a factor of 2.125.
In everyday lifeHelium has large zero-point vibrations and does not solidify at ordinary pressures, no matter how much it is cooled.
Detailed summary
Substituting the potential energy of a spring into the Schrödinger equation gives equally spaced energy levels ħω(n + ½), with a minimum energy of ½ħω. This is the lowest energy allowed by the uncertainty principle, which prevents position and momentum from both having definite values at the same time. Narrowing or broadening the wave function increases either the kinetic or the potential energy, raising the average energy. Meanwhile, the average position and momentum evolve according to the classical equation of motion, so classical mechanics survives in the averages of quantum mechanics.