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Angular Momentum and Spin

Martin Bojowald A

Published May 8, 2026• 0 citations

Abstract

If we include all three spatial dimensions, the time-dependent wave function https://www.w3.org/1998/Math/MathML" display="inline"> Ψ ( x , y , z , t ) depends on four coordinates. A wave function of this form determines the probability density https://www.w3.org/1998/Math/MathML" display="inline"> ρ t ( x , y , z ) = | Ψ ( x , y , z , t ) | 2 for measurements of the spatial position https://www.w3.org/1998/Math/MathML" display="inline"> ( x , y , z ) . The product https://www.w3.org/1998/Math/MathML" display="inline"> | Ψ ( x , y , z , t ) | 2 d x d y d z therefore equals the probability of finding a particle in a small cubic volume with side lengths https://www.w3.org/1998/Math/MathML" display="inline"> ( d x , d y , d z ) around https://www.w3.org/1998/Math/MathML" display="inline"> ( x , y , z ) at time t . The normalization condition for this probability density is given by 4.1 https://www.w3.org/1998/Math/MathML" display="block"> ∫ − ∞ ∞ ∫ − ∞ ∞ ∫ − ∞ ∞ | Ψ ( x , y , z , t ) | 2 d x d y d z = 1 for all t .

DOI: 10.1201/9781003588702-5

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