Quantum mechanics is a physical science dealing with the behavior of matter and energy on the scale of atoms and subatomic particles / waves.

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1 Quantum mechanics is a physical science dealing with the behavior of matter and energy on the scale of atoms and subatomic particles / waves. It also forms the basis for the contemporary understanding of how very large objects such as stars and galaxies, and cosmological events such as the Big Bang, can be analyzed and explained. The acceptance by the general physics community of quantum mechanics is due to its accurate prediction of the physical behavior of systems, including systems where Newtonian mechanics fails. Through a century of experimentation and applied science, quantum mechanical theory has proven to be very successful and practical. Wave function, in quantum mechanics, variable quantity that mathematically describes the wave characteristics of a particle. The value of the wave function of a particle at a given point of space and time is related to the likelihood of the particle s being there at the time. In other word, A wave function in quantum mechanics is a description of the quantum state of a system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements made on the system can be derived from it. 1

2 The wave function of a single particle, like its probability cloud, assigns an amplitude to all possible positions of the particle. Wave function is an assignment of a complex number to every point in space, at each time. The particle does not occupy a definite position at each time; instead, it is assigned a probability cloud that extends over all space. The density of the probability cloud at a point represents the relative probability of finding the particle at that point. Thus, the particle is more likely to be found where the density of the its probability cloud is high, and it is less likely to be found where the cloud s density is low. Quantum entanglement is a phenomenon where two particles are quantumly linked to each other regardless of how far apart they are. Disturbing one of the particles also disturbs the other. This principle has been used to encrypt information as any attempt to intercept one of the particles will disturb the other, which can then be detected. Quantum computing uses the property that quantum particles can exist in multiple states at the same time so can be used to carry out many calculations in parallel. Currently, very small quantum computers have been created, but at present there are technical difficulties involved in building bigger systems. 2

3 In quantum mechanics, the Schrödinger equation is a partial differential equation that describes how the quantum state of a quantum system changes with time. It was formulated in late 1925, and published in 1926, by the Austrian physicist Erwin Schrödinger. In classical mechanics, Newton's second law (F = ma) is used to make a mathematical prediction as to what path a given system will take following a set of known initial conditions. In quantum mechanics, the analogue of Newton's law is Schrödinger's equation for a quantum system (usually atoms, molecules, and subatomic particles whether free, bound, or localized). It is not a simple algebraic equation, but in general a linear partial differential equation, describing the time-evolution of the system's wave. Ĥ is the Hamiltonian operator (which characterizes the total energy of any given wave function and takes different forms depending on the situation). 3

4 There is a general equation that describes this wave-like behavior and, with the appropriate potential energy and boundary conditions, will predict the results of the experiments. The equation is called the Schrodinger equation and it forms the foundations of quantum theory. As a fundamental equation, Schrodinger s has been found to successfully predict every observable physical phenomenon at the atomic scale. Without this equation, we will not be able to understand the properties of electronic materials and the principles of operation of many semiconductor devices. 4

5 In three Dimension: These wavefunctions are called the Eigenfunctions (characteristic functions) of the system, and they determine the behavior and energy of the electron under steadystate conditions. The eigenfunctions are also called stationary states. 5

6 The first boundary condition is that ψ must be continuous. The second is that dψ /dx must be continuous. 6

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8 Inserting 1 st boundary condition: 8

9 Inserting 2nd boundary condition: 9

10 Normalization 10

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