Guided reading
The Born interpretation turns the wave function into experimental predictions. Once \(|\Psi|^2\) is accepted as probability density, normalization and probability current become necessary consistency conditions.
Born's interpretation
The wave function itself is complex, so it is not directly a probability. The measurable probability density is the squared modulus:
\[\rho(x,t)=|\Psi(x,t)|^2=\Psi^*(x,t)\Psi(x,t).\]
The probability of finding the particle in a small interval \(dx\) around \(x\) is \(dP=\rho(x,t)\,dx\).
Probability in an interval
\[\mathrm{Prob}(a\lt x\lt b)=\int_a^b |\Psi(x,t)|^2\,dx.\]
For a single particle somewhere on the line, the total probability must be one:
\[\int_{-\infty}^{+\infty}|\Psi(x,t)|^2\,dx=1.\]
Continuity equation
Probability is conserved locally. In one dimension, the density and probability current satisfy
\[\frac{\partial\rho}{\partial t}+\frac{\partial j}{\partial x}=0,\qquad j=\frac{\hbar}{2mi}\left(\Psi^*\frac{\partial\Psi}{\partial x}-\Psi\frac{\partial\Psi^*}{\partial x}\right).\]
The current \(j\) tells how probability flows across position, just as a fluid current tells how mass flows through space.
Conceptual checkpoint
- \(\Psi\) carries amplitude and phase information.
- \(|\Psi|^2\) is real and nonnegative.
- Normalization makes the wave function predictive.
- The continuity equation protects total probability during time evolution.
Challenge: derive the continuity equation
Challenge (see the book for solution). Starting from the TDSE and its complex conjugate, derive the continuity equation instead of taking \(\rho\) and \(j\) as definitions.
\[i\hbar\frac{\partial\Psi}{\partial t}=-\frac{\hbar^2}{2m}\frac{\partial^2\Psi}{\partial x^2}+V\Psi,\qquad -i\hbar\frac{\partial\Psi^*}{\partial t}=-\frac{\hbar^2}{2m}\frac{\partial^2\Psi^*}{\partial x^2}+V\Psi^*.\]
Multiply the first equation by \(\Psi^*\), the second by \(\Psi\), subtract the two expressions and rearrange the result as
\[\frac{\partial}{\partial t}(\Psi^*\Psi)+\frac{\partial}{\partial x}\left[\frac{\hbar}{2mi}\left(\Psi^*\frac{\partial\Psi}{\partial x}-\Psi\frac{\partial\Psi^*}{\partial x}\right)\right]=0.\]
This verifies \(\rho=\Psi^*\Psi\) and \(j=\frac{\hbar}{2mi}\left(\Psi^*\partial_x\Psi-\Psi\partial_x\Psi^*\right)\).
Conservation logic
The probability density tells how likely the particle is to be found near \(x\), while the current describes how that probability flows. The two must satisfy a local conservation law:
\[\rho=\Psi^*\Psi,\qquad \frac{\partial\rho}{\partial t}+\frac{\partial j}{\partial x}=0.\]
Integrating \(\rho\) over an interval gives the probability of finding the particle there. If probability leaves the interval, the current through the boundaries accounts for it.
Exercises can ask students to verify definitions or interpret signs of \(j\), while the full derivation from the TDSE remains a book-level calculation.
Exercise-ready boundary
This page is designed to support short guided exercises on: Born interpretation, continuity equation, normalization and probability in a spatial interval.
- Use from this page: the definitions, physical setup, highlighted equations and conceptual links needed to start a first calculation or explanation.
- Keep in the book: complete derivations, extended historical discussion, worked solutions and the full textbook narrative remain in the original chapter and linked book resources.
- Good exercise balance: ask the student to identify assumptions, apply one relation, and interpret the result physically, without requiring material not introduced on this page.
Practice anchors
Use these anchors to design compact exercises. The exercise should be answerable from this page plus standard algebra, while longer derivations, full worked examples and broader context should point back to the original book.
- Focus: Born interpretation, continuity equation, normalization and probability in a spatial interval.
- Conceptual check: state what the main result says physically before using it algebraically.
- Equation: \[\rho(x,t)=|\Psi(x,t)|^2=\Psi^*(x,t)\Psi(x,t).\]
- Equation: \[\mathrm{Prob}(a\lt x\lt b)=\int_a^b |\Psi(x,t)|^2\,dx.\]
- Equation: \[\int_{-\infty}^{+\infty}|\Psi(x,t)|^2\,dx=1.\]
- Boundary: use this page for setup and first-step reasoning; cite the book for longer derivations, complete experimental history or solved-problem detail.
- Typical task: derive, interpret, or apply the relation above to a simple case without introducing topics outside this page.
Original book and previews:
Source note: Original auxiliary summary for this book-app, based on Chapter 2 of Mario Reis, Quantum Mechanics, Elsevier, 2026. Book text and figures are copyright © 2026 Elsevier Inc. No original book figure is reproduced on this page.