University Quantum Mechanics I · The Schrödinger Equation · 2.03
Separation of variables and the constant E
Try Ψ(x, t) = ψ(x) f(t) and divide by it: each side then depends on one variable alone, so both equal a constant, and f(t) = exp(−iEt/ℏ) follows.
Course-map guide · not a complete lesson or simulationScope & orientation
What this subsection covers
Try Ψ(x, t) = ψ(x) f(t) and divide by it: each side then depends on one variable alone, so both equal a constant, and f(t) = exp(−iEt/ℏ) follows.A strong response uses level ladders beside their eigenfunctions and states where the model stops being reliable.
Reasoning checklist
Evidence, assumptions and limits
Assumptions to state
State the system, observable, approximation, and conditions held fixed before using a model.
Evidence to collect
Does a numerically evolved packet hold its norm constant, and does the probability lost from an interval match the current integrated through its two edges?
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Does a numerically evolved packet hold its norm constant, and does the probability lost from an interval match the current integrated through its two edges? Useful evidence includes ψ(x, t) from a stated integrator and grid, total norm against time, interval norm compared with the integrated flux at both edges, the drift traced to the time step, and the box edge named as the leak..
Limits to state
This GioPhysics course map is an adaptable learning sequence, not academic credit, accreditation, or a universal university syllabus.
Read the complete note
This GioPhysics course map is an adaptable learning sequence, not academic credit, accreditation, or a universal university syllabus. It is a first dedicated quantum-mechanics course, distinct from University Physics V, which covers quantum mechanics alongside atomic, nuclear and particle physics in twenty units: this course is narrower, slower, and teaches the linear algebra it needs rather than assuming it. Hydrogen appears here as an introduction, with the full radial derivation belonging to a second course. A midterm examination is assumed around week 8. Follow your institution's published scope, notation, laboratory programme, and assessment rules. A result should be checked against units, signs, limiting cases, and the conditions under which its model was derived.
Diagram & examples
Work the claim before choosing an equation
Interactive concept map
Follow the model from claim to evidence.
Try Ψ(x, t) = ψ(x) f(t) and divide by it: each side then depends on one variable alone, so both equal a constant, and f(t) = exp(−iEt/ℏ) follows. The trick needs V independent of t, and it returns special solutions, not the general one.
Model, evidence, and boundary
What must be stated before selecting an equation for Separation of variables and the constant E?
modelModel, evidence, and boundary turns the stated idea into a representation that can make a prediction.
Example questions
Try the reasoning before revealing the structure.
What must be stated before selecting an equation for Separation of variables and the constant E?
Quick check
Test the reasoning, not recall
What must be stated before selecting an equation for Separation of variables and the constant E?
Choose an answer to test the model.