page_title: Introduction to quantum mechanics
section_title: Application to the hydrogen atom
hierarchical_section_title: Introduction to quantum mechanics / Copenhagen interpretation / Application to the hydrogen atom
caption_attribution_description: English: Collection of 16 atomic single-electron orbitals for the lowest four n quantum numbers. Only the cases n = l + 1 {\displaystyle n=l+1} with one radial node are shown. Each row shows one l-quantum number l=0 (s), l=1 (p), l=2 (d) and l=3 (f) with m quantum numbers m &#x2208; { &#x2212; l , &#x2026; , l } {\displaystyle m\in \{-l,\ldots ,l\}} in ascending order. The color depicts the complex-valued phase of the wavefunction. In contrast to the real-valued orbitals in chemistry, which are superpositions of several m-eigenstates, the probability densities shown here are symmetric around the z-axis. The solid bodies enclose the volume where the continuous probability density exceeds a well-chosen threshold. The formulas that describe the orbitals can be found at Hydrogen atom. The labels in the image show the quantum numbers nlm with l encoded as letter.
context_page_description: Quantum mechanics is the science of the very small. It explains the behavior of matter and its interactions with energy on the scale of atomic and subatomic particles. By contrast, classical physics explains matter and energy only on a scale familiar to human experience, including the behavior of astronomical bodies such as the Moon. Classical physics is still used in much of modern science and technology. However, towards the end of the 19th century, scientists discovered phenomena in both the large and the small worlds that classical physics could not explain. The desire to resolve inconsistencies between observed phenomena and classical theory led to two major revolutions in physics that created a shift in the original scientific paradigm: the theory of relativity and the development of quantum mechanics. It describes these concepts in roughly the order in which they were first discovered.
  Light behaves in some aspects like particles and in other aspects like waves. Matter—the "stuff" of the universe consisting of particles such as electrons and atoms—exhibits wavelike behavior too.
context_section_description: Bohr's model of the atom was essentially a planetary one, with the electrons orbiting around the nuclear "sun".  However, the uncertainty principle states that an electron cannot simultaneously have an exact location and velocity in the way that a planet does.  Instead of classical orbits, electrons are said to inhabit atomic orbitals. An orbital is the "cloud" of possible locations in which an electron might be found, a distribution of probabilities rather than a precise location. Each orbital is three dimensional, rather than the two dimensional orbit, and is often depicted as a three-dimensional region within which there is a 95 percent probability of finding the electron.
  Schrödinger was able to calculate the energy levels of hydrogen by treating a hydrogen atom's electron as a wave, represented by the "wave function" Ψ, in an electric potential well, V, created by the proton. The solutions to Schrödinger's equation are distributions of probabilities for electron positions and locations. Orbitals have a range of different shapes in three dimensions.  The energies of the different orbitals can be calculated, and they accurately match the energy levels of the Bohr model.
  Within Schrödinger's picture, each electron has four properties:
  An "orbital" designation, indicating whether the particle wave is one that is closer to the nucleus with less energy or one that is farther from the nucleus with more energy;
  The "shape" of the orbital, spherical or otherwise;
  The "inclination" of the orbital, determining the magnetic moment of the orbital around the z-axis.
  The "spin" of the electron.
  
  The collective name for these properties is the quantum state of the electron. The quantum state can be described by giving a number to each of these properties; these are known as the electron's quantum numbers. The quantum state of the electron is described by its wave function. The Pauli exclusion principle demands that no two electrons within an atom may have the same values of all four numbers. 
  The first property describing the orbital is the principal quantum number, n, which is the same as in Bohr's model. n denotes the energy level of each orbital.  The possible values for n are integers:
  The next quantum number, the azimuthal quantum number, denoted l, describes the shape of the orbital. The shape is a consequence of the angular momentum of the orbital. The angular momentum represents the resistance of a spinning object to speeding up or slowing down under the influence of external force. The azimuthal quantum number represents the orbital angular momentum of an electron around its nucleus. The possible values for l are integers from 0 to n − 1 (where n is the principal quantum number of the electron):
  The shape of each orbital is usually referred to by a letter, rather than by its azimuthal quantum number. The first shape (l=0) is denoted by the letter s (a mnemonic being "sphere"). The next shape is denoted by the letter p and has the form of a dumbbell. The other orbitals have more complicated shapes (see atomic orbital), and are denoted by the letters d, f, g, etc.
  The third quantum number, the magnetic quantum number, describes the magnetic moment of the electron, and is denoted by mₗ (or simply m).  The possible values for mₗ are integers from −l to l (where l is the azimuthal quantum number of the electron):
  The magnetic quantum number measures the component of the angular momentum in a particular direction.  The choice of direction is arbitrary; conventionally the z-direction is chosen.
  The fourth quantum number, the spin quantum number (pertaining to the "orientation" of the electron's spin) is denoted mₛ, with values +​¹⁄₂ or −​¹⁄₂.
  The chemist Linus Pauling wrote, by way of example:
  In the case of a helium atom with two electrons in the 1s orbital, the Pauli Exclusion Principle requires that the two electrons differ in the value of one quantum number. Their values of n, l, and mₗ are the same. Accordingly they must differ in the value of mₛ, which can have the value of +​¹⁄₂ for one electron and −​¹⁄₂ for the other."
  I
caption: The shapes of atomic orbitals. Rows: 1s, 2p, 3d and 4f. From left to right 
    
      
        
          m
          =
          −
          l
          ,
          …
          ,
          l
        
      
      {\displaystyle m=-l,\ldots ,l}
    
  . The colors show the phase of the wave function.
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