However, though the ideas of classical physics explain interference and diffraction phenomena relating to the propagation of light, they do not account for the absorption and emission of light. All bodies radiate electromagnetic energy as heat; in fact, a body emits radiation at all wavelengths. The energy radiated at different wavelengths is a maximum at a wavelength that depends on the temperature of the body; the hotter the body, the shorter the wavelength for maximum radiation. Attempts to calculate the energy distribution for the radiation from a blackbody using classical ideas were unsuccessful.
A blackbody is a hypothetical ideal body or surface that absorbs and reemits all radiant energy falling on it. One formula, proposed by Wilhelm Wien of Germany, did not agree with observations at long wavelengths, and another, proposed by Lord Rayleigh John William Strutt of England, disagreed with those at short wavelengths.
In the German theoretical physicist Max Planck made a bold suggestion.
First-principles quantum chemistry in the life sciences.
He assumed that the radiation energy is emitted, not continuously, but rather in discrete packets called quanta. Planck showed that the calculated energy spectrum then agreed with observation over the entire wavelength range. Quantum mechanics. Article Media. Info Print Print. Table Of Contents. Submit Feedback. Thank you for your feedback.
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More About. Articles from Britannica Encyclopedias for elementary and high school students. Consider the integral. Evidently, the Hamiltonian is a hermitian operator. It is postulated that all quantum-mechanical operators that represent dynamical variables are hermitian. The sets of energies and wavefunctions obtained by solving any quantum-mechanical problem can be summarized symbolically as solutions of the eigenvalue equation.
The result is. This is quite reasonable from a physical point of view since eigenvalues represent possible results of measurement.
Thus eigenfunctions belonging to different eigenvalues are orthogonal. But it is always possible to construct degenerate functions that are mutually orthogonal. A general result is therefore the orthonormalization condition. It is easy to prove that a linear combination of degenerate eigenfunctions is itself an eigenfunction of the same energy. Consider now the linear combination. There is evidently a limitless number of possible eigenfunctions for a degenerate eigenvalue. However, only d of these will be linearly independent.
The Principles of Quantum Mechanics by Paul A.M. Dirac
The term orthogonal has been used both for perpendicular vectors and for functions whose product integrates to zero. This actually connotes a deep connection between vectors and functions. Consider two orthogonal vectors a and b. Then, in terms of their x, y, z components, labeled by 1, 2, 3, respectively, the scalar product can be written. Suppose now that we consider an analogous relationship involving vectors in n -dimensional space which you need not visualize!
We could then write. Finally let the dimension of the space become non-denumerably infinite, turning into a continuum. But this is just the relation for orthogonal functions. A function can therefore be regarded as an abstract vector in a higher-dimensional continuum, known as Hilbert space. This is true for eigenfunctions as well. The integral over the product of the two functions is then analogous to a scalar product or inner product in linear algebra of the abstract vectors, written.
Eq 6 ] is therefore expressed as. One of the extraordinary features of quantum mechanics is the possibility for superpositions of states. The state of a system can sometimes exist as a linear combination of other states, for example,. Note that the expectation value need not itself be a possible result of a single measurement like the centroid of a donut, which is located in the hole! We say that the operators do not commute.
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This can be attributed to the perturbing effect one measurement on a quantum system can have on subsequent measurements. An example of non-commuting operators from everyday life. In our usual routine each morning, we shower and we get dressed.
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But the result of carrying out these operations in reversed order will be dramatically different!