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University Quantum Mechanics I · The Schrödinger Equation · 2.06

Probability density and probability current

ρ(x, t) = |Ψ(x, t)|², a density of dimension one over length, and j(x, t) = (ℏ/m) Im(Ψ* dPsi/dx), of dimension one over time - the rate at which probability flows past x.

Course-map guide · not a complete lesson or simulation

Scope & orientation

What this subsection covers

ρ(x, t) = |Ψ(x, t)|², a density of dimension one over length, and j(x, t) = (ℏ/m) Im(Ψ* dPsi/dx), of dimension one over time - the rate at which probability flows past x.

A strong response uses psi and psi' matched across a finite potential step and states where the model stops being reliable.

Reasoning checklist

Evidence, assumptions and limits

01

Assumptions to state

State the system, observable, approximation, and conditions held fixed before using a model.

02

Evidence to collect

Do a clamped string's resonances sit at the eigenvalues of the same boundary-value problem that quantises a particle in a box, and where does the analogy fail?

Read the complete note

Do a clamped string's resonances sit at the eigenvalues of the same boundary-value problem that quantises a particle in a box, and where does the analogy fail? Useful evidence includes resonant frequencies against mode number with a linear fit and residuals, end correction and stiffness as systematics, and the disanalogy stated: one kₙ = n π / L feeds two different dispersion relations, so f goes as n here and E as n squared there..

03

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.

01 · Physical claimProbability density and probability current

ρ(x, t) = |Ψ(x, t)|², a density of dimension one over length, and j(x, t) = (ℏ/m) Im(Ψ* dPsi/dx), of dimension one over time - the rate at which probability flows past x.

Read the complete note

ρ(x, t) = |Ψ(x, t)|², a density of dimension one over length, and j(x, t) = (ℏ/m) Im(Ψ* dPsi/dx), of dimension one over time - the rate at which probability flows past x. Both are built from Psi alone, and j vanishes wherever psi is real, so a bound stationary state carries no current.

02 · RepresentationThe subsection's claim

Model, evidence, and boundary

03 · TestPrediction before measurement

Which response about Probability density and probability current is most defensible?

04 · Evidence & boundaryDecide, then qualify

Do a clamped string's resonances sit at the eigenvalues of the same boundary-value problem that quantises a particle in a box, and where does the analogy fail?

Read the complete note

Do a clamped string's resonances sit at the eigenvalues of the same boundary-value problem that quantises a particle in a box, and where does the analogy fail? Useful evidence includes resonant frequencies against mode number with a linear fit and residuals, end correction and stiffness as systematics, and the disanalogy stated: one kₙ = n π / L feeds two different dispersion relations, so f goes as n here and E as n squared there..

Interactive diagram for Probability density and probability current: follow the physical claim through its representation, proposed test, evidence, and model boundary.

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.

Diagram check

Which response about Probability density and probability current is most defensible?

Quick check

Test the reasoning, not recall

Which response about Probability density and probability current is most defensible?

Choose an answer to test the model.