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CRC Press eBooks

Examples

Martin Bojowald A

Published May 8, 2026• 0 citations
Publisher

Abstract

The Schrödinger equation 2.1 https://www.w3.org/1998/Math/MathML" display="block"> i ℏ ∂ Ψ ∂ t = − ℏ 2 2 m ∂ 2 Ψ ∂ x 2 + V ( x ) Ψ
DOI
10.1201/9781003588702-3

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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 .

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The unknowns that remain in quantum mechanics are not the mysteries that people often think of. The real mysteries of quantum mechanics are annexed to foundational knowledge of quantum physics that, given all indications, just appeared into existence at our measurement level, or second layer of quantum physics. These miracles of knowledge include Planck's energy equation https://www.w3.org/1998/Math/MathML" display="inline"> E = h ν , Born's rule https://www.w3.org/1998/Math/MathML" display="inline"> | ψ 〉 | ψ 〉 ∗ = | | ψ 〉 | 2 , elements of the Schrödinger equation, and Dirac's notation. In Quantum Clear we elucidate the physics leading to https://www.w3.org/1998/Math/MathML" display="inline"> E = h ν and https://www.w3.org/1998/Math/MathML" display="inline"> | ψ 〉 | ψ 〉 ∗ = | | ψ 〉 | 2 , but that's it. And it remains certain that those who introduced this wonderful knowledge did not elaborate further except for the explanation of Schrödinger who linked mathematical discovery to ‘like a great gift…’. Ward, on the other hand, described the arrival of knowledge with ‘ no way I can trace the source. ’

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There is more information in a quantum state than in the classical initial values https://www.w3.org/1998/Math/MathML" display="inline"> ( x 0 , p 0 ) that determine the complete 1-dimensional motion of a point particle for a given force. The initial wave function https://www.w3.org/1998/Math/MathML" display="inline"> Ψ ( x , 0 ) requires an infinite number of parameters, which can be expressed in observational terms as the two basic expectation values https://www.w3.org/1998/Math/MathML" display="inline"> ⟨ x ^ ⟩ and https://www.w3.org/1998/Math/MathML" display="inline"> ⟨ p ^ ⟩ together with the two variances https://www.w3.org/1998/Math/MathML" display="inline"> ( σ x 2 , σ p 2 ) and the covariance 5.1 https://www.w3.org/1998/Math/MathML" display="block"> C x p = 1 2 ⟨ x ^ p ^ + p ^ x ^ ⟩ − ⟨ x ^ ⟩ ⟨ p ^ ⟩ ,

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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 .
The unknowns that remain in quantum mechanics are not the mysteries that people often think of. The real mysteries of quantum mechanics are annexed to foundational knowledge of quantum physics that, given all indications, just appeared into existence at our measurement level, or second layer of quantum physics. These miracles of knowledge include Planck's energy equation https://www.w3.org/1998/Math/MathML" display="inline"> E = h ν , Born's rule https://www.w3.org/1998/Math/MathML" display="inline"> | ψ 〉 | ψ 〉 ∗ = | | ψ 〉 | 2 , elements of the Schrödinger equation, and Dirac's notation. In Quantum Clear we elucidate the physics leading to https://www.w3.org/1998/Math/MathML" display="inline"> E = h ν and https://www.w3.org/1998/Math/MathML" display="inline"> | ψ 〉 | ψ 〉 ∗ = | | ψ 〉 | 2 , but that's it. And it remains certain that those who introduced this wonderful knowledge did not elaborate further except for the explanation of Schrödinger who linked mathematical discovery to ‘like a great gift…’. Ward, on the other hand, described the arrival of knowledge with ‘ no way I can trace the source. ’
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