## Section 10.6: Two-State Superpositions

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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

Ψ_{n1n2}(*x*,*t*) = 2^{−1/2} [ψ_{n1}(*x*,*t*) + ψ_{n2}(*x*,*t*)] ,

and

Φ_{n1n2}(*p*,*t*) = 2^{−1/2} [φ_{n1}(*p*,*t*) + φ_{n2}(*p*,*t*)] ,

where ψ_{n}(*x*,*t*) = exp(−*i*E_{n}*t*/*ħ*) ψ_{n}(*x*) and φ_{n2}(*p*,*t*) = exp(−*iE*_{n}*t*/*ħ*) φ_{n}(*p*). We can write these wave functions in a way that stresses their relative phases:

Ψ_{n1n2}(*x*,*t*) = 2^{−1/2} exp(−*iE*_{n1}*t*/*ħ*) [ψ_{n1}(*x*,*t*) + exp(−*i*(*E*_{n2} − *E*_{n1})*t*/*ħ*) ψ_{n2 }(*x*,*t*)] , (10.11)

and

Φ_{n1n2}(*p*,*t*) = 2^{−1/2} exp(−*iE*_{n1}*t*/*ħ*) [φ_{n1}(*p*,*t*) + exp(−*i*(*E*_{n2} − *E*_{n1})*t*/*ħ*) φ_{n2}(*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.*, Δ*t* = 1 corresponds to an elapsed time of 2π*ħ*/*E*_{1}. You can change *n*_{1} and *n*_{2}, the default values, *n*_{1} = 1 and *n*_{2} = 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 *n*_{1} and *n*_{2}.

^{5}One 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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