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Mathematics of the Slater Determinant

This post, largely based on Reference 1, is about the Slater determinant. The Slater determinant is related to the Hartree Product. In that post, it was determined that a Hartree Product does not meet the antisymmetry principle. The Slater determinant is one way to fix this problem. John C. Slater, developer of the Slater determinant. Two Electrons…

Hartree Product

In quantum chemistry, a wave function for a single electron is called an orbital or a spin orbital, but what could be a wave function for several electrons? [1] One option is related to the Hartree Product. Douglas Hartree, a contributor to the field of quantum chemistry. Noninteracting System According to Attila Szabo and Neil…

Notes on Section 1, The need for a quantum theory

Paul Dirac wrote a book called “The Principles of Quantum Mechanics.” It is a very logical book with later sections depending on earlier sections. I have only read bits and pieces of this book, but now I going to embark on the challenging task of reading it chronologically while keeping a critical eye. In addition…

Development of the Slater Determinant

Introduction What is a Slater Determinant, and how was it developed? At least part of the answer can be found in a book called ‘Modern Quantum Chemistry’ [1]. In order to address this question it is necessary to study the Pauli Exclusion Principle and the antisymmetry principle. The following content is based on chapter 2…

The Origin of the Ket in Quantum Mechanics

Origin of the Ket The origin of the ket in quantum mechanics arises from the need to describe the concept of superposition as well as a system that 1) changes with respect to time and 2) has so-called states. The ket can symbolize such a state. The concept of superposition is closely related to the…

Gradient of a Function

Introduction In this post, I find an expression for the gradient of a function, in terms of spherical coordinates. This is a continuation of previous posts, such as this one. This post has a lot of symbols, but there is a lot of repetition. Formalism Recall that $ \vec{e}_r = \frac{\partial x(r)}{\partial r}\big|_{r^+} \vec{e}_x + \frac{\partial y(r)}{\partial…

Relating Unit Vectors to a Jacobian Matrix

In this post, I relate coefficients of unit vectors to derivatives and to a Jacobian matrix that was used in a previous post. Unit Vectors Three unit vectors for a right-handed spherical coordinate system are $  \vec{e}_r = \sin \theta \cos\phi \vec{e}_x + \sin \theta \sin\phi \vec{e}_y + \cos \theta \vec{e}_z $ $  \vec{e}_{\theta} =  \cos\theta…

The Radial Unit Vector

In this post, I find an expression for the radial unit vector, $\vec{e}_r$. The three unit vectors in the following digram form a right-handed spherical coordinate system. This unit vector is easier to find than the other two unit vectors because all that is needed is vector addition. The Radial Unit Vector in Terms of…

The Polar Unit Vector

Consider a spherical coordinate system. Let a point be represented by $(r, \theta, \phi)$, in that order. Now that the order of the coordinates is established, I can define unit vectors that form a right-handed coordinate system. Suppose the radial unit vector $\vec{e}_r$ points radially outward from the origin to the point, and the polar…

The Azimuthal Unit Vector

In this post, I write the azimuthal unit vector $\vec{e}_{\phi}$ in terms of Cartesian coordinates. Here, $\phi$ is the azimuthal angle in the $x-y$ plane. As noted in this post, $\vec{e}_{\phi}$ points in the direction of increasing $\phi$. Geometrical Setup Since $\vec{e}_{\phi}$ is perpendicular to the line segment from the origin to the point $(x,y,0)$, I am…

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