Final answer:
The wave function ψ(z,t) is found by solving Schrödinger's time-independent equation for the spatial part and then multiplying the result by a time-modulation factor. The general form is Ψ(x, t) = Aei(kx-ωt), where the magnitudes of this function give real, measurable quantities.
Step-by-step explanation:
The student asked how to find both the position and time dependence of the wave function ψ(z,t). To find this, we need to consider two equations from quantum mechanics: Schrödinger's time-independent equation and Schrödinger's time-dependent equation. The time-independent solution is typically found first, and then multiplied by a time-modulation factor to achieve the full time-dependent solution.
The general form of the time-dependent wave function for a quantum particle can be written as:
Ψ(x, t) = Aei(kx-ωt)
where A is the amplitude, k is the wave number, and ω is the angular frequency. The term ei(kx-ωt) incorporates the wave's spatial and temporal dependence and despite involving an imaginary number, leads to real, measurable quantities when the magnitudes of the wave function are squared.
To answer the student's question, we would use the time-independent wave function, often designated as ψ(x)—the solution of the Schrödinger's time-independent equation—and multiply it by the appropriate time-dependent factor φ(t) to find Ψ(x, t).