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  The Schrodinger Equation, Physics tutorial

Introduction:

The fact that the matter behaves similar to a wave implies that we would require an equation or a set of equations to explain the wave behavior of matter. As such, we would anticipate having an equation explaining the wave nature of the electron. Similarly, the quantum-mechanical treatment of the harmonic oscillator would occupy an equation explaining the wave behavior of the oscillator. Here, we shall derive the suitable differential equations for the wave-function, and as a result, explain the probability by which the particle would be found at various points  in the suitable area of confinement, and possibly beyond.

The Schrodinger Equation:

At the starting of the 20th century, experimental evidence recommended that the atomic particles were as well wave-like in nature. For illustration, electrons were found to give diffraction patterns if passed via a double slit in a similar manner to light waves. Thus, it was reasonable to suppose that a wave equation could describe the behavior of atomic particles.

Schrodinger was the very first person to introduce such a wave equation. Much conversation then centered on what the equation signify. The Eigen-value of the wave equation were represented to be equivalent to the energy levels of the quantum mechanical system, and the best test of the equation was when it was employed to resolve for the energy levels of the Hydrogen atom, and the energy levels were found to be in accordance with the Rydberg's Law.

This was initially much less apparent what the wave-function of the equation was. After much debate, the wave-function is now accepted to be the probability distribution. The Schrodinger equation is employed to determine the allowed energy levels of the quantum mechanical systems (like atoms, or transistors). The related wave function provides the probability of determining the particle at a certain position.

Derivation:

We are quite conscious that the total mechanical energy of a body is the sum of the kinetic energy 'T' and the potential energy 'V'. Obviously, you as well remember that the kinetic energy of a body of mass 'm' is:

(1/2) mv2 = (1/2) (m2v2/m) = p2/2m

Then, the sum of its kinetic and potential energies, that is, the net or total mechanical energy is:

E = T + V = (p2/2m) + V

You surely remind that we can write p = (h/2π)k and E = (h/2π)ω, replacing these into equation, we get:

(h/2π) ω = [(h2/4π2) k2]/2m + V

Let us assume that the case of a free particle (V = 0). Then,

(h/2π) ω = (h2k2/8π2m) + V

Let us try,

Ψ(x, t) = Aei (kx - ωt)

Then,

(∂2/∂x2) Ψ(x,t) = - k2 Aei(kx - ωt) = - k2 Ψ(x,t)

(∂/∂t) Ψ(x,t) = - iω Aei(kx - ωt) = -iω Ψ(x,t)

Multiplying the first equation via - h2/8π2m and the second by i(h/2π)

By computing and rearranging we get,

(∂2Ψ/∂x2) + (8π2m/h2) (E - V) Ψ = 0

Here,

(∂2/∂x2) = Second derivative with respect to x

Ψ = Schrodinger wave function

E = Energy

V = Potential Energy

The answer to this equation is a wave which explains the quantum features of a system. Though, physically interpreting the wave is one of the major philosophical problems of quantum mechanics.

The solution to the equation is mainly based on the process of Eigen Values devised through Fourier. This is where any mathematical function is deduced as the sum of an infinite series of other periodic functions. The trick is to determine the correct functions that encompass the right amplitudes so that whenever added altogether through superposition they give the desired solution.

Therefore, the solution to Schrodinger's equation, the wave function for the system, was substituted through the wave functions of the individual series, natural harmonics of each other, and an infinite series. Schrodinger has invented that the replacement waves illustrated the individual states of the quantum system and their amplitudes provide the relative significance of that state to the whole system.

Schrodinger's equation represents the entire wave such as the properties of matter and was one of greatest accomplishments of 20th century science.

This is used in physics and most of chemistry to deal by problems regarding the atomic structure of matter. This is a very powerful mathematical tool and the whole basis of the wave mechanics.

Schrodinger equation is the name of the fundamental non-relativistic wave equation employed in one version of quantum mechanics to explain the behavior of a particle in a field of force. There is the time dependant equation employed for explaining progressive waves, applicable to the motion of the free particles. And the time independent form of this equation employed for illustrating the standing waves.

Schrodinger's time-independent equation can be resolved logically for a number of simple systems. The time-dependant equation is of the first order in time however of the second order by respect to the co-ordinates, therefore it is not consistent by relativity. The solutions for bound systems provide three quantum numbers, corresponding to the three co-ordinates, and an estimated relativistic correction is possible by comprising the fourth spin quantum number.

Interpretation of the equation and its solutions:

1) Usually, the solution Ψ(x,t) of this equation is usually a complex function. We are already familiar that the magnitude of a complex function might not make any sense physically. Though the square of the magnitude (that is, the intensity) certainly does, being for all time a positive real number. However, it is the measurement of the probability of the particle being illustrated arriving somewhere. |Ψ(x,t)|2 is, thus, a probability density. Obviously, this implies that, |Ψ(x,t)|2 dxis the probability of determining the particle between x and x + dx  at time t.

2) The solution for the above equation for a given functions V(x) are possible for numerous solutions, each having a corresponding energy 'E'. This makes sure that only certain values of 'E' are allowed (that is, energy quantization).

3) For a system in which the particle is in one of such levels, 

Ψ(x,t) = Ψ(x) e-iEt/(h/2π)

Then, |Ψ(x,t)|2 = |Ψ(x)| as |e-iEt/(2π)|2 = 1 and the probability of determining the particle at any point is independent of the time. Such a state is termed as a stationary state.

4) Any function that satisfies the time-independent Schrodinger equation should be (i) single valued (ii) continuous; (iii) smoothly varying; and (iv) tend to zero as

x → ± ∞

5) The probability of determining the particle all along the chosen 1-dimension is 1, Therefore,

-∞∫∞ |Ψ(x)|2 dx = 1

This is the normalization condition, and it deduces the certainty (probability 1) of determining the particle anywhere in the range

-∞ < x < ∞

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