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Section 10.6: Two-State Superpositions

n1 = | n2 =

check, then click the input values and play button to see the probability densities instead.

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One of the simplest examples of non-trivial time-dependent states is that of an equal mix, two-state superposition in the infinite square well.5  The position- and momentum-space wave functions are just

Ψn1n2(x,t) = 2−1/2n1(x,t) + ψn2(x,t)] ,


Φn1n2(p,t) = 2−1/2n1(p,t) + φn2(p,t)] ,

where ψn(x,t) = exp(−iEnt/ħ) ψn(x) and φn2(p,t) = exp(−iEnt/ħ) φn(p). We can write these wave functions in a way that stresses their relative phases:

Ψn1n2(x,t) = 2−1/2 exp(−iEn1t/ħ) [ψn1(x,t) + exp(−i(En2En1)t/ħ) ψn2 (x,t)] , (10.11)


Φn1n2(p,t) = 2−1/2 exp(−iEn1t/ħ) [φn1(p,t) + exp(−i(En2En1)t/ħ) φn2(p,t)] . (10.12)

In this case there is a time-dependent relative phase that depends on the difference in energy eigenvalues and an overall time-dependent phase in Eqs. (10.11) and (10.12).

The animation depicts the time dependence of an arbitrary equal-mix two-state superposition.   Restart. The time is given in terms of the time it takes the ground-state wave function to return to its original phase, i.e., Δt = 1 corresponds to an elapsed time of 2πħ/E1. You can change n1 and n2, the default values, n1 = 1 and n2 = 2 represent the standard case treated in almost every textbook. Explore the time-dependent form of the position-space and momentum-space wave functions for other n1 and n2.

5One of the earliest pedagogical visualizations of the time dependence of such a two-state system is by C. Dean, "Simple Schrödinger Wave Functions Which Simulate Classical Radiating Systems," Am. J. Phys. 27, 161-163 (1959).

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