jtbell said:The angular part of an orbital wave function for hydrogen (or any other spherically symmetric potential) is a spherical harmonic:
[tex]\Psi_{nlm}(r,\theta,\phi) = R_{nl}(r)Y_{lm}(\theta,\phi)[/tex]
The angular parts look the same, because they are identical (see jtbell's comment). Unlike spherical harmonics, orbital wave functions, however, do not consist only of an angular part. They also have a radial part. And this radial part is non-trival and comes from solving the Schroedinger equation for some potential (e.g., in hydrogen the nuclear attraction of the proton, in higher spherical atoms from nuclear attraction and the mean field of the other electrons (Fock potential)). But this has no influence on the angular part. E.g., 2p and 3p orbitals have the same angular part, not only in a single atom, but across all atoms (in the nonrelativistic case etc.).Chemist20 said:yes but for a 2p for example, what's the difference in representation between the orbital itself and the spherical harmonic? they look the same to me.
cgk said:The angular parts look the same, because they are identical (see jtbell's comment). Unlike spherical harmonics, orbital wave functions, however, do not consist only of an angular part. They also have a radial part. And this radial part is non-trival and comes from solving the Schroedinger equation for some potential (e.g., in hydrogen the nuclear attraction of the proton, in higher spherical atoms from nuclear attraction and the mean field of the other electrons (Fock potential)). But this has no influence on the angular part. E.g., 2p and 3p orbitals have the same angular part, not only in a single atom, but across all atoms (in the nonrelativistic case etc.).
Note that multiplying the radial wave function by constant factor changes the size, not the shape, of the "drawing" of the orbital, which is really just a drawing of the surface of maximum probability density. By looking at how more general changes in the radial wave function affects this surface, you can see why spherical harmonics look so much like these surfaces.Chemist20 said:yes, but when drawing the 2p orbital and the 2p spherical harmonic what's the difference? THEY ARE THE SAME!
Spherical harmonics are mathematical functions that describe the spatial distribution of a wave in three dimensions. They are used to represent the angular component of wavefunctions in quantum mechanics. Wavefunctions, on the other hand, are complex-valued functions that describe the quantum state of a physical system.
Spherical harmonics and wavefunctions are crucial in understanding the behavior of particles at the quantum level. They are used to solve the Schrödinger equation, which is the fundamental equation in quantum mechanics. They also provide insight into the energy levels and properties of atoms, molecules, and other quantum systems.
Spherical harmonics are a type of wavefunction that describes the angular component of a wave. They are used in conjunction with other types of wavefunctions, such as radial wavefunctions, to fully describe the quantum state of a system. The product of the two types of wavefunctions gives the overall wavefunction for a system.
The physical interpretations of spherical harmonics and wavefunctions vary depending on the context. In general, they represent the probability amplitude of finding a particle in a specific energy state and location. Spherical harmonics, specifically, describe the shape of electron orbitals in an atom.
Spherical harmonics and wavefunctions are used extensively in a wide range of fields, including physics, chemistry, and engineering. In physics, they are used to model atomic and molecular systems, while in engineering they are used to analyze and design antennas and other wave-based technologies. Additionally, they are also used in computer graphics and image processing to represent and manipulate 3D data.