# Angular Momentum In Spherical Coordinates Pdf

3.2.4 Eulerian Equations Angular Momentum Theorem in a. angular momentum which is called spin and typically is denoted S~. Spin is a new degree Spin is a new degree of freedom in addition to the spacial coordinates (x,y,z)., Summary of Angular Momentum C. David Sherrill School of Chemistry and Biochemistry Georgia Institute of Technology November 1999 We start from the classical expression for angular momentum, L = r p, to obtain the.

### 19. Spherical Coordinates physics.weber.edu

Theory of Angular Momentum and Spin University Of Illinois. Because the angular momentum operators derived above correspond to three‐dimensional rotations, it is natural to seek their eigenfunctions in spherical coordinates. After the transformation from Cartesian, •In spherical coordinates, the Laplacian is given by: angular momentum •In coordinate representation, the eigenvalue equation for L z becomes: •Which has the solution: •The wave-function must be single-valued, so that: •Integer m values occur only for integer l values •Therefore half-integer l values are forbidden for the case of orbital angular momentum ihθ,φl,mhmθ,φl,m φ.

it is convenient to express the angular momentum operators in spherical polar coordinates: r,θ,φ, rather than the Cartesian coordinates x, y, z. The spherical coordinates are related Orbital Angular Momentum Operators in Spherical Coordinates The standard angular momentum basis is an eigenbasis of the operators (L 2 ,L z ), with certain phase and other conventions.

Angular momentum algebra It is easy to see that the operator J2 = J xJ x +J yJ y +J zJ z commutes with the operators J x, J y and J z, [J2,J i] = 0. We choose the component J in spherical polar coordinates The mathematical discussion of the properties of the angular momentum of a particle becomes convenient if it is expressed in spherical polar coordinates r, θ, ϕ .

Angular momentum algebra It is easy to see that the operator J2 = J xJ x +J yJ y +J zJ z commutes with the operators J x, J y and J z, [J2,J i] = 0. We choose the component J These are exactly the angular momentum quantum number and magnetic quantum number, respectively, that are mentioned in General Chemistry classes. If we consider spectroscopic notation, an angular momentum quantum number of zero suggests that we have an s orbital if all of $$\psi(r,\theta,\phi)$$ is present.

• The Wave Function: The Schrödinger Equation, Statistical Interpretation, Momentum, The Uncertainty Principle, Hilbert Space, Operators and Observables, Dirac Notation • Time-Independent Schrödinger Equation : Stationary States, Simple Exactly Solvable Appendix A. The Angular Momentum Operator in Spherical Coordinates The transformation from cartesian to spherical coordinates for the angular mo­

angular momentum which is called spin and typically is denoted S~. Spin is a new degree Spin is a new degree of freedom in addition to the spacial coordinates (x,y,z). Atomic and Molecular Quantum Theory Course Number: C561 15 Spherical Harmonics 1. We will now derive the eigenstates of the total angular momentum and z-component of

Orbital angular momentum and the spherical harmonics March 28, 2013 1 Orbital angular momentum as the z-component of the angular momentum vector for the mass (L z ), so Equation 3 states that angular momentum is conserved in the zdirection for our spherical pendulum.

Atomic and Molecular Quantum Theory Course Number: C561 15 Spherical Harmonics 1. We will now derive the eigenstates of the total angular momentum and z-component of previous index next PDF. Orbital Angular Momentum in Three Dimensions. Michael Fowler, UVa. The Angular Momentum Operators in Spherical Polar Coordinates

1 Raising and lowering operators for angular momentum quantum numbers l in spherical harmonics Q. H. Liu∗, D. M. Xun, and L. Shan Key Laboratory for Micro-Nano Optoelectronic Devices of Ministry of Education, and State Key The classical angular momentum operator is orthogonal to both lr and p as it is built from the cross product of these two vectors. Happily, these properties also hold for the quantum angular

### Chapter 5. Angular Momentum Inside Mines

Lecture 4 Angular Momentum The Hydrogen Atom Recall. 3 Orbital angular momentum 3.1 Orbital angular momentum in spherical coordinates We use here the usual spherical coordinate system (ˆ; ;’), with associated basis vectors e, the angular momentum in the guided coordinate system (GCS). The second term of the left The second term of the left hand side is the change of angular momentum due to the rotation of the coordinate system..

AngularMomentum.pdf Spin (Physics) Angular Momentum. Physics 505 Homework No. 5 Solutions S5-1 1. Angular momentum uncertainty relations. A system is in the lmeigenstate of L2, Lz. (a) Show that the expectation values of …, Orbital Angular Momentum Operators in Spherical Coordinates The standard angular momentum basis is an eigenbasis of the operators (L 2 ,L z ), with certain phase and other conventions..

### Quantum Mechanics ch9 Angular momentum IOPscience

B. Commutation relations of L Condensed Matter Physics. 1 Raising and lowering operators for angular momentum quantum numbers l in spherical harmonics Q. H. Liu∗, D. M. Xun, and L. Shan Key Laboratory for Micro-Nano Optoelectronic Devices of Ministry of Education, and State Key Indeed, working in spherical polar coordinates (r; ;’), with the polar axis along the cartesian zdirection we get L^ x = i h sin’ @ @ +cot cos’ @ @’! L^ y = i h cos’ @ @ +cot sin’ @ @’! (5) L^ z = i h @ @’ We can therefore conclude that the angular momentum operators commute with the Hamilto-nian of a particle in a central eld, for example a Coulomb eld, hence L^2 and L^ z, say.

These are exactly the angular momentum quantum number and magnetic quantum number, respectively, that are mentioned in General Chemistry classes. If we consider spectroscopic notation, an angular momentum quantum number of zero suggests that we have an s orbital if all of $$\psi(r,\theta,\phi)$$ is present. 1.1. ORBITAL ANGULAR MOMENTUM - SPHERICAL HARMONICS 3 Since J+ raises the eigenvalue m by one unit, and J¡ lowers it by one unit, these operators are referred to as raising and lowering operators, respectively.

98 Theory of Angular Momentum and Spin Properties of Rotations in R 3 Rotational transformations of vectors ~r2R 3, in Cartesian coordinates ~r= (x Angular momentum theory is used in a large number of applications in chemical physics. Some examples are: atomic orbital theory, rotational spectra, many electron theory of atoms, NMR and ESR spectroscopy, spin-orbit coupling.

of this angular momentum vector in a cartesian coordinate system located at the origin mentioned above are given in terms of the cartesian coordinates of r and p as follows: L z = x p y - y p x, L x = y p z - z py, L y = z p x - x p z. Using the fact that the quantum mechanical coordinate operators {q k} = x, y, z as well as the conjugate momentum operators {p j} = p x, py, pz are Hermitian 5.61 Spherical Harmonics page 1 ANGULAR MOMENTUM Now that we have obtained the general eigenvalue relations for angular momentum directly from …

Angular momentum operators - preview We will have operators corresponding to angular momentum about different orthogonal axes , , and though they will not commute with one another in contrast to the linear momentum operators for the different coordinate directions, , and which do commute ˆ L x ˆ L y ˆ L z ˆ p x ˆ p y ˆ p z. Angular momentum operators - preview We will, however, find The alternative is to realize that in any problem with spherical symmetry we ex- pect the solutions to have a physical interpretation in terms of angular momentum. Recall that in classical mechanics, when a particle moves under the in

Spherical Polar Coordinates The motion of a free particle on the surface of a sphere will involve components of angular momentum in three-dimensional space. Spherical polar coordinates provide the most convenient description for this and related problems with spherical symmetry. 1.1. ORBITAL ANGULAR MOMENTUM - SPHERICAL HARMONICS 3 Since J+ raises the eigenvalue m by one unit, and J¡ lowers it by one unit, these operators are referred to as raising and lowering operators, respectively.

Orbital angular momentum and the spherical harmonics April 12, 2017 1 Orbital angular momentum Orbital Angular Momentum Operators in Spherical Coordinates The standard angular momentum basis is an eigenbasis of the operators (L 2 ,L z ), with certain phase and other conventions.

to its spherical coordinates equivalent with equations (6.3). A derivation of how this is A derivation of how this is done is beyond the scope of our analysis, and we will simply insert the corresponding Using spherical coordinates the kinetic energy is T = 1 2 m 2 θ˙2 + ϕ˙2 sin2 θ), (4) and, with a downward directed z-axis, see ﬁgure 1, the potential energy is V = mg(1 −cosθ). (5) The zero point of V is chosen so that the equilibrium, vertically downwards, of the pendulum corresponds to T = V = 0. The energy, E = T + V, is conserved. The z-component of the angular momentum, L z

The alternative is to realize that in any problem with spherical symmetry we ex- pect the solutions to have a physical interpretation in terms of angular momentum. Recall that in classical mechanics, when a particle moves under the in • The Wave Function: The Schrödinger Equation, Statistical Interpretation, Momentum, The Uncertainty Principle, Hilbert Space, Operators and Observables, Dirac Notation • Time-Independent Schrödinger Equation : Stationary States, Simple Exactly Solvable

of this angular momentum vector in a cartesian coordinate system located at the origin mentioned above are given in terms of the cartesian coordinates of r and p as follows: L z = x p y - y p x, L x = y p z - z py, L y = z p x - x p z. Using the fact that the quantum mechanical coordinate operators {q k} = x, y, z as well as the conjugate momentum operators {p j} = p x, py, pz are Hermitian Physics 216 Spring 2012 Clebsch-Gordon coeﬃcients and the tensor spherical harmonics Consider a system with orbital angular momentum L~ and spin angular momentum ~S.

If looking for the ebook Physiology of Sport and Exercise, Fourth Edition by Dr. David Costill, Dr. Jack H. Wilmore in pdf format, then you have come on to right website. Physiology of sport and exercise wilmore pdf Nicholls Rivulet Book Review: Physiology of Sport and Exercise, 3rd edition. Jack H Wilmore PhD and David L Costill PhD. Champaign, Illinois: Human Kinetics. 2004. Hard cover, illustrated, 726 pages, \$75. Jack H Wilmore PhD and David L Costill PhD.

## Angular Momentum Operator Identities G

Spherical Harmonics Chemistry LibreTexts. Angular momentum plays a central role in discussing central potentials, i.e. potentials that only depend on the radial coordinate r. It will also prove useful to have expression, 5.61 Spherical Harmonics page 1 ANGULAR MOMENTUM Now that we have obtained the general eigenvalue relations for angular momentum directly from ….

### Chapter 2 Angular Momentum Hydrogen Atom and Helium Atom

Lecture 11 { Spin orbital and total angular momentum 1. 3 Orbital angular momentum 3.1 Orbital angular momentum in spherical coordinates We use here the usual spherical coordinate system (ˆ; ;’), with associated basis vectors e, it is convenient to express the angular momentum operators in spherical polar coordinates: r,θ,φ, rather than the Cartesian coordinates x, y, z. The spherical coordinates are related.

it is convenient to express the angular momentum operators in spherical polar coordinates: r,θ,φ, rather than the Cartesian coordinates x, y, z. The spherical coordinates are related Eigenfunctions of Orbital Angular Momentum In Cartesian coordinates, the three components of orbital angular momentum can be written (364) using the Schrödinger representation. Transforming to standard spherical polar coordinates,

as the z-component of the angular momentum vector for the mass (L z ), so Equation 3 states that angular momentum is conserved in the zdirection for our spherical pendulum. Appendix A. The Angular Momentum Operator in Spherical Coordinates The transformation from cartesian to spherical coordinates for the angular mo­

the angular momentum in the guided coordinate system (GCS). The second term of the left The second term of the left hand side is the change of angular momentum due to the rotation of the coordinate system. of this angular momentum vector in a cartesian coordinate system located at the origin mentioned above are given in terms of the cartesian coordinates of r and p as follows: L z = x p y - y p x, L x = y p z - z py, L y = z p x - x p z. Using the fact that the quantum mechanical coordinate operators {q k} = x, y, z as well as the conjugate momentum operators {p j} = p x, py, pz are Hermitian

24CHAPTER2. ANGULARMOMENTUM,HYDROGENATOM,ANDHELIUMATOM 2.1 Angular momentum and addition of two an-gular momenta 2.1.1 Schr odinger Equation in 3D to its spherical coordinates equivalent with equations (6.3). A derivation of how this is A derivation of how this is done is beyond the scope of our analysis, and we will simply insert the corresponding

Orbital angular momentum and the spherical harmonics March 28, 2013 1 Orbital angular momentum Summary of Angular Momentum C. David Sherrill School of Chemistry and Biochemistry Georgia Institute of Technology November 1999 We start from the classical expression for angular momentum, L = r p, to obtain the

Because the angular momentum operators derived above correspond to three‐dimensional rotations, it is natural to seek their eigenfunctions in spherical coordinates. After the transformation from Cartesian Orbital Angular Momentum Operators in Spherical Coordinates The standard angular momentum basis is an eigenbasis of the operators (L 2 ,L z ), with certain phase and other conventions.

Indeed, working in spherical polar coordinates (r; ;’), with the polar axis along the cartesian zdirection we get L^ x = i h sin’ @ @ +cot cos’ @ @’! L^ y = i h cos’ @ @ +cot sin’ @ @’! (5) L^ z = i h @ @’ We can therefore conclude that the angular momentum operators commute with the Hamilto-nian of a particle in a central eld, for example a Coulomb eld, hence L^2 and L^ z, say Hamiltonian is expressed in spherical coordinates, and the angular momentum operator L ˆ 2 re- placed with its exact quantum eigenvalues, then the one-dimensional radial Schro

Overview. The Kerr metric is a generalization of the Schwarzschild metric, discovered by Karl Schwarzschild in 1915, which described the geometry of spacetime around an uncharged, spherically-symmetric, and non-rotating body. Physics 505 Homework No. 5 Solutions S5-1 1. Angular momentum uncertainty relations. A system is in the lmeigenstate of L2, Lz. (a) Show that the expectation values of …

Angular momentum operators - preview We will have operators corresponding to angular momentum about different orthogonal axes , , and though they will not commute with one another in contrast to the linear momentum operators for the different coordinate directions, , and which do commute ˆ L x ˆ L y ˆ L z ˆ p x ˆ p y ˆ p z. Angular momentum operators - preview We will, however, find 56 Chapter 5. Angular Momentum Review: If the potential has spherical symmetry, V(r)=V(r), e.g., Coulomb potential V(r)=Z 1 Z 2 e2/r, coordinate separation can be performed using the spherical coordinates

Orbital angular momentum and the spherical harmonics April 12, 2017 1 Orbital angular momentum Summary of Angular Momentum C. David Sherrill School of Chemistry and Biochemistry Georgia Institute of Technology November 1999 We start from the classical expression for angular momentum, L = r p, to obtain the

Angular momentum theory is used in a large number of applications in chemical physics. Some examples are: atomic orbital theory, rotational spectra, many electron theory of atoms, NMR and ESR spectroscopy, spin-orbit coupling. 56 Chapter 5. Angular Momentum Review: If the potential has spherical symmetry, V(r)=V(r), e.g., Coulomb potential V(r)=Z 1 Z 2 e2/r, coordinate separation can be performed using the spherical coordinates

These are exactly the angular momentum quantum number and magnetic quantum number, respectively, that are mentioned in General Chemistry classes. If we consider spectroscopic notation, an angular momentum quantum number of zero suggests that we have an s orbital if all of $$\psi(r,\theta,\phi)$$ is present. Angular momentum and spherical harmonics. The angular part of the Laplace operator can be written: (12.1) The angular part of the Laplacian is related to the angular momentum of a wave in quantum theory. In units where , the angular momentum operator is: (12.4) and (12.5) Note that in all of these expressions , etc. are all operators. This means that they are applied to the functions on

56 Chapter 5. Angular Momentum Review: If the potential has spherical symmetry, V(r)=V(r), e.g., Coulomb potential V(r)=Z 1 Z 2 e2/r, coordinate separation can be performed using the spherical coordinates denote the spherical coordinates parameterising the unit sphere (see ﬁgure). Previously, we obtained the eigenvalues of the angular momentum operator by making use of the raising and lowering operators in a manner that parallelled

Spherical Polar Coordinates The motion of a free particle on the surface of a sphere will involve components of angular momentum in three-dimensional space. Spherical polar coordinates provide the most convenient description for this and related problems with spherical symmetry. denote the spherical coordinates parameterising the unit sphere (see ﬁgure). Previously, we obtained the eigenvalues of the angular momentum operator by making use of the raising and lowering operators in a manner that parallelled

47 E. Ladder operators Our next job is to construct what in the jargon is known as the ’representation’ of the angular momentum operators. By this we mean - understand how we can write the angular momentum operators as ﬁnite-size matrices. of this angular momentum vector in a cartesian coordinate system located at the origin mentioned above are given in terms of the cartesian coordinates of r and p as follows: L z = x p y - y p x, L x = y p z - z py, L y = z p x - x p z. Using the fact that the quantum mechanical coordinate operators {q k} = x, y, z as well as the conjugate momentum operators {p j} = p x, py, pz are Hermitian

Hamiltonian is expressed in spherical coordinates, and the angular momentum operator L ˆ 2 re- placed with its exact quantum eigenvalues, then the one-dimensional radial Schro Overview. The Kerr metric is a generalization of the Schwarzschild metric, discovered by Karl Schwarzschild in 1915, which described the geometry of spacetime around an uncharged, spherically-symmetric, and non-rotating body.

in spherical polar coordinates The mathematical discussion of the properties of the angular momentum of a particle becomes convenient if it is expressed in spherical polar coordinates r, θ, ϕ . Lecture 7: Angular Momentum, Hydrogen Atom 1 Vector Quantization of Angular Momen-tum and Normalization of 3D Rigid Rotor wavefunctions Consider l= 1, so L2 = 2~2.

### Summary angular momentum derivation

Lecture 4 Angular Momentum The Hydrogen Atom Recall. Eigenfunctions of Orbital Angular Momentum In Cartesian coordinates, the three components of orbital angular momentum can be written (364) using the Schrödinger representation. Transforming to standard spherical polar coordinates,, Angular momentum in spherical coordinates Peter Haggstrom www.gotohaggstrom.com mathsatbondibeach@gmail.com December 6, 2015 1 Introduction Angular momentum is a deep property and in courses on quantum mechanics a lot of.

### CHM 532 Notes on Angular Momentum Eigenvalues and

Summary angular momentum derivation. The alternative is to realize that in any problem with spherical symmetry we ex- pect the solutions to have a physical interpretation in terms of angular momentum. Recall that in classical mechanics, when a particle moves under the in 98 Theory of Angular Momentum and Spin Properties of Rotations in R 3 Rotational transformations of vectors ~r2R 3, in Cartesian coordinates ~r= (x.

• Angular momentum University of Babylon
• 19. Spherical Coordinates physics.weber.edu

• it is convenient to express the angular momentum operators in spherical polar coordinates: r,θ,φ, rather than the Cartesian coordinates x, y, z. The spherical coordinates are related angular momentum which is called spin and typically is denoted S~. Spin is a new degree Spin is a new degree of freedom in addition to the spacial coordinates (x,y,z).

Because the angular momentum operators derived above correspond to three‐dimensional rotations, it is natural to seek their eigenfunctions in spherical coordinates. After the transformation from Cartesian as the z-component of the angular momentum vector for the mass (L z ), so Equation 3 states that angular momentum is conserved in the zdirection for our spherical pendulum.

Eigenfunctions of Orbital Angular Momentum In Cartesian coordinates, the three components of orbital angular momentum can be written (364) using the Schrödinger representation. Transforming to standard spherical polar coordinates, Summary of Angular Momentum C. David Sherrill School of Chemistry and Biochemistry Georgia Institute of Technology November 1999 We start from the classical expression for angular momentum, L = r p, to obtain the

B.3. ANGULAR MOMENTUM IN SPHERICAL COORDINATES B.3 Angular Momentum in Spherical Coordinates The orbital angular momentum operator Z can be expressed in spherical coordinates … to its spherical coordinates equivalent with equations (6.3). A derivation of how this is A derivation of how this is done is beyond the scope of our analysis, and we will simply insert the corresponding

56 Chapter 5. Angular Momentum Review: If the potential has spherical symmetry, V(r)=V(r), e.g., Coulomb potential V(r)=Z 1 Z 2 e2/r, coordinate separation can be performed using the spherical coordinates the angular momentum in the guided coordinate system (GCS). The second term of the left The second term of the left hand side is the change of angular momentum due to the rotation of the coordinate system.

ANGULAR MOMENTUM So far, we have studied simple models in which a particle is subjected to a force in one dimension (particle in a box, harmonic oscillator) or forces in three dimensions (particle in a 3-dimensional box). We were able to write the Laplacian, ∇2, in terms of Cartesian coordinates, assuming ψ to be a product of 1-dimensional wavefunctions. By separation of variables, we were 56 Chapter 5. Angular Momentum Review: If the potential has spherical symmetry, V(r)=V(r), e.g., Coulomb potential V(r)=Z 1 Z 2 e2/r, coordinate separation can be performed using the spherical coordinates

Indeed, working in spherical polar coordinates (r; ;’), with the polar axis along the cartesian zdirection we get L^ x = i h sin’ @ @ +cot cos’ @ @’! L^ y = i h cos’ @ @ +cot sin’ @ @’! (5) L^ z = i h @ @’ We can therefore conclude that the angular momentum operators commute with the Hamilto-nian of a particle in a central eld, for example a Coulomb eld, hence L^2 and L^ z, say • The Wave Function: The Schrödinger Equation, Statistical Interpretation, Momentum, The Uncertainty Principle, Hilbert Space, Operators and Observables, Dirac Notation • Time-Independent Schrödinger Equation : Stationary States, Simple Exactly Solvable

• The Wave Function: The Schrödinger Equation, Statistical Interpretation, Momentum, The Uncertainty Principle, Hilbert Space, Operators and Observables, Dirac Notation • Time-Independent Schrödinger Equation : Stationary States, Simple Exactly Solvable 98 Theory of Angular Momentum and Spin Properties of Rotations in R 3 Rotational transformations of vectors ~r2R 3, in Cartesian coordinates ~r= (x

j2 ).220) ~u ~ ~ e¡i®J¢^ = e¡i®(J1 +J2 )¢^u (5. j1 . j2 . a doubly-degenerate state associated with a two-fold degenerate j = 1=2 state is a doublet. to the description of a single particle with spin (for which J~ = L ~ +S ~ is the sum of the orbital and spin angular momenta of the particle). for this combined space the total angular momentum vector is the sum J~ = J~1 + J~2 (5.218) form j2 ).220) ~u ~ ~ e¡i®J¢^ = e¡i®(J1 +J2 )¢^u (5. j1 . j2 . a doubly-degenerate state associated with a two-fold degenerate j = 1=2 state is a doublet. to the description of a single particle with spin (for which J~ = L ~ +S ~ is the sum of the orbital and spin angular momenta of the particle). for this combined space the total angular momentum vector is the sum J~ = J~1 + J~2 (5.218) form

Angular momentum plays a central role in discussing central potentials, i.e. potentials that only depend on the radial coordinate r. It will also prove useful to have expression Orbital Angular Momentum Operators in Spherical Coordinates The standard angular momentum basis is an eigenbasis of the operators (L 2 ,L z ), with certain phase and other conventions.

Overview. The Kerr metric is a generalization of the Schwarzschild metric, discovered by Karl Schwarzschild in 1915, which described the geometry of spacetime around an uncharged, spherically-symmetric, and non-rotating body. Using spherical coordinates the kinetic energy is T = 1 2 m 2 θ˙2 + ϕ˙2 sin2 θ), (4) and, with a downward directed z-axis, see ﬁgure 1, the potential energy is V = mg(1 −cosθ). (5) The zero point of V is chosen so that the equilibrium, vertically downwards, of the pendulum corresponds to T = V = 0. The energy, E = T + V, is conserved. The z-component of the angular momentum, L z

1.2 Alternative coordinate systems 3 α z = z y y0 1 x x r 0 1 0 1 Figure 1.2: The frame S1 is obtained from 0 by a rotation around their common z axis. and y1(t) diﬀer from x0(t) and y0(t). The reason for this is the spin angular momentum. l We will see in the next section how this comes about.1) which are called fundamental commutation relations for angular momentum.. for spin angular momentum. . a+ ˆ ˆ operator analytical description of the simple harmonic oscillator in that it is an algebraic approach. This approach is a little similar to the approach. rather than a or an

The reason for this is the spin angular momentum. l We will see in the next section how this comes about.1) which are called fundamental commutation relations for angular momentum.. for spin angular momentum. . a+ ˆ ˆ operator analytical description of the simple harmonic oscillator in that it is an algebraic approach. This approach is a little similar to the approach. rather than a or an Orbital angular momentum and the spherical harmonics March 28, 2013 1 Orbital angular momentum

Angular momentum plays a central role in discussing central potentials, i.e. potentials that only depend on the radial coordinate r. It will also prove useful to have expression Angular momentum plays a central role in discussing central potentials, i.e. potentials that only depend on the radial coordinate r. It will also prove useful to have expression