Relation between energy and frequency of a quanta of radiation
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- Energy:
![{\displaystyle E}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b)
- Frequency:
![{\displaystyle f}](https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61)
- Angular Frequency:
![{\displaystyle \omega =2\pi f}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7a1bf35d395c2d52391265e4bbda0aed14f52579)
- Wavenumber:
![{\displaystyle k=2\pi /\lambda }](https://wikimedia.org/api/rest_v1/media/math/render/svg/eecfda2366cb3a64ef6d4747e2460284155a1960)
- Plank's Constant:
![{\displaystyle h}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b26be3e694314bc90c3215047e4a2010c6ee184a)
- Wavelength:
![{\displaystyle \lambda }](https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a)
- Momentum:
![{\displaystyle p}](https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36)
Phase of a Plane Wave Expressed as a Complex Phase Factor
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- Ket:
![{\displaystyle |\psi (t)\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/fe09d6c91bfdbae7a3aaa7f0ae7ff6b96f521eca)
- Reduced Planck's Constant:
![{\displaystyle \hbar }](https://wikimedia.org/api/rest_v1/media/math/render/svg/de68de3a92517953436c93b5a76461d49160cc41)
- Hamiltonian:
![{\displaystyle H(t)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9d1b6c8837aed2794e7b52afa88ad371f1d275fa)
- The Hamiltonian describes the total energy of the system.
- Partial Derivative:
![{\displaystyle \partial /\partial t}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b92814eb5d3b151bdb7851482e0704b34053dfdd)
- Mass:
![{\displaystyle m}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc)
- Potential:
![{\displaystyle V}](https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845)
Begin with a step from the time-independent derivation
Set each side equal to a constant,
Multiply by
to remove constants on the right side of the equation.
The solution is similar to what was found previously
The amplitude at a point
is equal to the amplitude at a point
The following equation must be true:
Rewrite
in terms of the frequency
Enter the equation into the expression of
The time-dependent Schrodinger equation is a product of two 'sub-functions'
To extract
, differentiate with respect to time:
Rearrange:
![{\displaystyle -{\frac {\hbar ^{2}}{2m}}{\frac {d^{2}\psi (x)}{dx^{2}}}+U(x)\psi (x)=E\psi (x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/742246936e20734d209115d990af1b02b69a6c27)
![{\displaystyle \left[-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}+U(\mathbf {r} )\right]\psi (\mathbf {r} )=E\psi (\mathbf {r} )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bdf858301e2a61ba8e87afde1f2929f2371f8ff6)
- Del Operator:
![{\displaystyle \nabla }](https://wikimedia.org/api/rest_v1/media/math/render/svg/a3d0e93b78c50237f9ea83d027e4ebbdaef354b2)
The Schrodinger Equation is based on two formulas:
- The classical wave function derived from the Newton's Second Law
- The de Broglie wave expression
Formula of a classical wave:
Separate the function into two variables:
Insert the function into the wave equation:
Rearrange to separate
and
Set each side equal to an arbitrary constant,
Solve this equation
The amplitude at one point needs to be equal to the amplitude at another point:
The following condition must be true:
Incorporate the de Broglie wave expression
Use the symbol
Use the expression of kinetic energy,
Modify the equation by adding a potential energy term and the Laplacian operator
In non-relativistic quantum mechanics, the Hamiltonian of a particle can be expressed as the sum of two operators, one corresponding to kinetic energy and the other to potential energy. The Hamiltonian of a particle with no electric charge and no spin in this case is:
- kinetic energy operator:
![{\displaystyle T={\frac {p^{2}}{2m}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4c21a7e69cb7bb43957796cca84706e2dd0e177d)
- mass of the particle:
![{\displaystyle m\;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/35649ba0e080d57035272cec57371ea991b018f5)
- momentum operator:
![{\displaystyle \mathbf {p} =-\mathrm {i} \hbar \nabla }](https://wikimedia.org/api/rest_v1/media/math/render/svg/021da3a166b3998602e267f5185481e5028cf65f)
- potential energy operator:
![{\displaystyle V=V\left(\mathbf {r} \right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e8eae1dcc9d261ffdc1195094fe3325917fddb06)
- real scalar function of the position operator
: ![{\displaystyle V}](https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845)
- Gradient operator:
![{\displaystyle \nabla }](https://wikimedia.org/api/rest_v1/media/math/render/svg/a3d0e93b78c50237f9ea83d027e4ebbdaef354b2)
- Laplace operator:
![{\displaystyle \nabla ^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f4be87ad083e5ead48d92b0c82f2d4e719cb34a6)