Quantized energy level of linear harmonic oscillator
Quantization of energy levels refers to the idea that energy levels in certain physical systems can only exist at specific discrete values, rather than being continuous. This concept was first introduced by Max Planck in 1900 as part of his work on blackbody radiation. Quantization of energy levels in bound states is a fundamental concept in quantum mechanics. It arises from the wave-particle duality of matter, where particles such as electrons are described by wavefunctions. In the context of bound states, like those found in atoms, molecules, or other potential wells, the quantization of energy levels is a result of the wave nature of particles.
The key idea behind quantization is that only certain discrete energy levels are allowed for a particle in a bound state. This is in contrast to classical physics, where energy is considered continuous. The quantization of energy levels is described by the Schrödinger equation, which is a fundamental equation in quantum mechanics.
The Schrödinger equation describes the behaviour of a quantum particle in terms of a wavefunction, which gives the probability amplitude of finding the particle in a particular state. The solutions to the equation yield a set of wavefunctions, each associated with a different energy level. Quantization of energy levels arises from the requirement that the wavefunction must satisfy certain boundary conditions. The energy levels allowed for a given system are discrete and quantized because only certain wavefunctions (eigenfunctions) satisfy these boundary conditions. It is a fundamental principle of quantum mechanics and has important implications for the behaviour and properties of quantum systems.
The Schrödinger equation for a non-relativistic particle in a potential well is given by:
Hψ=Eψ
Here, H is the Hamiltonian operator, ψ is the wavefunction, and E is the energy of the system. The solutions to this equation provide the possible wavefunctions and corresponding energy levels for a given potential.
The quantization of energy levels is a consequence of the boundary conditions imposed by the potential well. The solutions to the Schrödinger equation must satisfy certain conditions at the boundaries of the potential well, leading to the quantization of energy levels.
For example, in the case of an electron bound to an atomic nucleus, the potential well is created by the attractive Coulomb force between the negatively charged electron and the positively charged nucleus. The solutions to the Schrödinger equation for this system result in quantized energy levels, which correspond to the allowed electron orbits in an atom.
These quantized energy levels are often visualized as electron shells or orbitals in atomic models. Each shell corresponds to a specific energy level, and electrons can only occupy these discrete energy levels.
In summary, the quantization of energy levels in bound states is a fundamental aspect of quantum mechanics, and it plays a crucial role in understanding the behaviour of particles in potential wells, such as electrons in atoms. For more clarification,
For region I
For region II
For region III
For region IV
But the wavefunction 𝜓_II and 𝜓_IV are in the same region, so they must be the same wavefunction and it can be different by some multiple of each other.
And N= (-1)^n where n= 0, 1, 2...
Since the integral between the turning points, ∫ p(x) dx is equal to the half of the closed integral over the complete cycle. ∲ p(x)dx, i.e
which is the Bohr Sommerfield quantization phase integral. It gives the WKB quantized energy levels for a bound state.
This condition was developed as an extension of Niels Bohr's model of the hydrogen atom, incorporating the de Broglie wavelength concept. Arnold Sommerfeld further refined Bohr's model by introducing the quantization condition in terms of an integral over the classical orbit.
The significance of the Bohr-Sommerfeld quantization condition lies in its attempt to bridge classical mechanics and quantum mechanics. Here are some key points regarding its significance:
Quantization of Energy Levels: The condition enforces the quantization of energy levels in a manner that goes beyond Bohr's original model. It introduces the idea that the classical action (integral of momentum) should be quantized, connecting the discrete energy levels of quantum mechanics with the classical orbits.
Wave-Particle Duality: The Bohr-Sommerfeld quantization condition is a step towards incorporating wave-particle duality into the understanding of atomic structure. By considering the de Broglie wavelength and the integral of the momentum over the orbit, it recognizes the wave nature of particles.
Transition to Quantum Mechanics: While the Bohr-Sommerfeld model has limitations and was later superseded by more sophisticated quantum mechanics, it provided a transitional framework that contributed to the development of quantum theory. The quantization condition served as an important step towards the formulation of modern quantum mechanics.
Limitations and Historical Context: The quantization condition was an attempt to extend classical mechanics to describe atomic phenomena. However, it had limitations, and its success was limited to certain simple systems. Ultimately, the full development of quantum mechanics, with the Schrödinger equation and wave functions, provided a more comprehensive and accurate description of atomic and subatomic behaviour.
Bohr's Correspondence Principle: The quantization condition embodies Bohr's correspondence principle, which states that quantum mechanics should reduce to classical mechanics in the limit of large quantum numbers. The quantization of action is consistent with this principle.
For a continuum of energy values of a semi classical system, only those energies En are allowed for which the area of contour are equal to (n+1/2)ℏ. Thus finally we write,
So, the area between two consecutive bound states in the phase space is
That means each single state in the phase space corresponds to an area h.
This note is a part of the Physics Repository.