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Question:

Write an essay on molecule.

 

Answer:

Every molecule is made up of atoms which are in turn composed of nuclei and electrons. When the behavior of nuclei and electrons were first being studied by the scientists the experimental findings were being tried to be interpreted in terms of Newtonian motions which eventually was unsuccessful. It was discovered that small particles of light do not behave in accordance to the Newtonian motions (Baker, 2014). It was observed that electrons as well as other small particles of light exhibit wave-like characteristics. After putting a lot of effort, a new theory was devised in 1924 which came to be known as quantum mechanics and currently is the fundamental framework for a pure understanding of sub-nuclear, nuclear and atomic physics along with condensed matter physics. The laws prior to the quantum mechanics are termed as classical mechanics. Even if the classical mechanics is considered to be an approximation to quantum mechanics, much of the framework of quantum theory has been inherited from the classical theory itself (Ball, 2014).

 

The emergence of quantum mechanics attempted to explain the phenomena stated below:

The quantum state

The idea that objects follow a trajectory path in 3D space should be abandoned especially for microscopic objects for 2 reasons;

  1. If a particle follows a trajectory, it should possess a particular momentum and position at every moment in time. Nonetheless, no experiment has been able to verify this theory.
  2. The assumption that electrons and photons move through a trajectory concludes something which cannot be in alignment with experiments (Bartels, 2011).

In classical theory, the 1D motion is represented by considering a particle which moves to and fro in a thin closed pipe which has a length L. The objection which is raised to this representation is that when a set of N orthogonal vectors are introduced, along with it an N-dimensional space is introduced which is composed of the linear combinations of the mentioned vectors (BaÅŸar, 2010). The motion of electron can be viewed as:

As suggested by Benioff, (2016) in a complex N-dimensional vector space unit vectors represent the physical state of an electron. In the space having N number of dimensions, the motion of the electron correlates with a trajectory which is along the surface of a unit sphere in the space. This signifies that the travel of electron can be smooth as well as continuous. In order to accomplish the new representation of motion, the limit ε → 0 is required to be taken where the number of intervals sent is N → ∞. Thus, the physical states are considered to be the vectors of unit length where the dimensional vector is infinite and is termed as Hilbert Space. Hilbert space can be defined as the vector space has infinite dimensions of all functions that are square-integrable (Tsang, 2013).

 

A set of generalized momentum and coordinates enumerates the physical state of a system in classical mechanics. Through time, a trajectory is traced by the physical state through the phase space. When a single particle has a 3 dimensional movement, the physical state is designated by {~x, p~} whereas the phase space is six dimensional. The trajectory in the 6D phase space is projected on the 3D subspace which is stretched through all the three axes (Hammerer, 2013).

The physical state of a moving point-like particle in 1D, in quantum mechanics, is identified by a wavefunction ψ(x,t) at every moment of time. At any given moment, the wavefunction is considered to be only a function of x and also as a vector in Hilbert space (Louko, 2011). Through time, the path is followed by the vector in Hilbert space. Therefore, if a rough analogy is made to the unit vector’s motion in spaces that have finite dimensions, it can be imagined that a path is traced by the tip of the vector on the unit sphere’s surface given that the space is infinite- dimension in this case (Lu, 2011).

Dynamics of quantum state

A trajectory present in 3D is used to represent the classical motion of a particle whereas; a curve on the surface of a unit sphere represents the quantum mechanical motion of a particle, provided that the space is infinite-dimensional. The question that arises here is how two theories can be so different since the classical theory is regarded as an approximation of quantum theory. The answer to this would be that the physical state of the particle is inconsequential. The thing which matters is the observation of the position of the particle which is in approximation of the classical trajectory ("Non-hermitean quantum mechanics", 2011). Even if the quantum state of a particle is not in alignment with a particular point in 3d space, but a trajectory is traced by the expectation value of a position of the particle through ordinary space in time t given by

< x >≡ Z dx xψ∗ (x,t)ψ(x,t)

The laws of motion < x > can be derived if the equation of motion for quantum state ψ(x,t), is provided and also can be compared to the classical laws of motion x(t). The most probable possibility is that both of the sets might end up looking similar which is termed as “Ehrenfest’s Principle”.

 

The Schrodinger wave equation

The number of dimensions is reliant on the number of particles as well as the number of spatial dimensions that are required in order for the characterization of the motion and position of the particle.

An electron is moving across a 2 dimensional space which has a mass m and charge, e which defines the x, y plane. It is assumed that a constant but not time varying potential is experienced by the electron at every point in the plane (Roser, 2016). The Schrodinger equation is given by       

  • ih (∂ψ/∂t) = −( h2 2 / 2m ∂x2) ψ + V (x)ψ

But ultimately, instead of the Schrodinger equation, the boundary conditions are responsible for the determination of discretion or continuity of the eigenvalues.

 

References

Baker, M. (2014). Lectures on quantum mechanics. Quantum Inf Process, 13(9), 2149-2151. https://dx.doi.org/10.1007/s11128-014-0796-9

Ball, P. (2014). Thermodynamics Confronts Quantum Mechanics. Physics, 7. https://dx.doi.org/10.1103/physics.7.35

Bartels, L. (2011). Visualizing quantum mechanics. Physics, 4. https://dx.doi.org/10.1103/physics.4.64

BaÅŸar, E. (2010). From “Quantum Mechanics to the Quantum Brain”. Neuroquantology, 8(3). https://dx.doi.org/10.14704/nq.2010.8.3.333

Benioff, P. (2016). Effects of a scalar scaling field on quantum mechanics. Quantum Inf Process. https://dx.doi.org/10.1007/s11128-016-1312-1

Hammerer, K. (2013). Quantum Mechanics Tackles Mechanics. Science, 342(6159), 702-703. https://dx.doi.org/10.1126/science.1245797

Louko, J. (2011). Advanced Mechanics and General Relativity. Class. Quantum Grav., 28(8), 089002. https://dx.doi.org/10.1088/0264-9381/28/8/089002

Lu, J. (2011). Transformation of Quantum States in Quantum Computation. AMM, 80-81, 276-278. https://dx.doi.org/10.4028/www.scientific.net/amm.80-81.276

Non-hermitean quantum mechanics. (2011). Int. J. Quantum Chem., 112(15), 2764-2765. https://dx.doi.org/10.1002/qua.23301

Roser, P. (2016). Quantum mechanics on York slices. Class. Quantum Grav., 33(6), 065001. https://dx.doi.org/10.1088/0264-9381/33/6/065001

Tsang, M. (2013). Testing quantum mechanics: a statistical approach. Quantum Measurements And Quantum Metrology, 1. https://dx.doi.org/10.2478/qmetro-2013-0007

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