What is Wave Function?
Quantum Physics or Quantum mechanics is a branch of science that deals with the study and behavior of matter as well as light. The wave function in quantum mechanics can be used to illustrate the wave properties of a particle. Therefore, a particle’s quantum state can be described using its wave function.
This interpretation of wave function helps define the probability of the quantum state of an element as a function of position, momentum, time, and spin. It is represented by a Greek alphabet Psi, 𝚿.
However, it is important to note that there is no physical significance of wave function itself. Nevertheless, its proportionate value of 𝚿2 at a given time and point of space does have physical importance.
Furthermore, we will discuss the Schrodinger equation, which was introduced in 1925 to define wave function.
Schrodinger Equation
In 1925, Erwin Schrodinger introduced this partial differential equation for wave function definition as a reward to the Quantum mechanics branch. According to him, the wave function can be satisfied and solved. Here is a time-dependent equation of Schrodinger shown in the image below.
ih \[\frac {∂}{∂t} 𝚿 (\overrightarrow{r}, t) = \frac{-h2 }{2m}∇2 + V(\overrightarrow{r}, t)] 𝚿 (\overrightarrow{r}, t) \]
In the above equations,
m refers to the particle’s mass.
∇ is laplacian.
h equals to h/2𝝿, which is also known as the reduced Planck’s constant.
i is the imaginary unit.
E is a constant matching energy level of a system
Properties of Wave Function
There must be a single value for 𝚿, and it must be continuous.
It is easy to compute the energy using the Schrodinger equation.
Wave function equation is used to establish probability distribution in 3D space.
If there is a particle, then the probability of finding it becomes 1.
Properties which can be measured for a particle should be known.
Normalization of Wave Function
In this scenario, the probability of finding a particle becomes 1 if it exists in the system. This depicts that the exact form of wave function 𝚿 is found.
Quantum Mechanics Postulates
It gets easier to decipher the force system wherein a particle in a conservative field resides with the help of a wave function.
Time independent Schrodinger’s equation was derived using the time-dependent equation.
The 6 Postulates of Quantum Mechanics are:
Associated with any particle moving in a conservative field of force is a wave function which determines everything that can be known about the system.
With every physical observable q there is associated an operator Q, which when operating upon the wave function associated with a definite value of that observable will yield that value times the wavefunction.
Any operator Q associated with a physically measurable property q will be Hermitian.
The set of eigenfunctions of operator Q will form a complete set of linearly independent functions.
Described by a given wave function for a system, the expected value of any property q can be found by performing the expectation value integral with respect to that wavefunction.
The time evolution of the wavefunction is given by the time dependent Schrodinger equation.
The total wavefunction must be antisymmetric with respect to the interchange of all coordinates of one fermion with those of another. Electronic spin must be included in this set of coordinates. The Pauli exclusion principle is a direct result of this antisymmetry principle.
The Physical Significance of Wave Function
There is no physical meaning of wave function as it is not a quantity which can be observed. Instead, it is complex. It is expressed as𝚿 (x, y, z, t) = a + ib and the complex conjugate of the wave function is expressed as𝚿 \[\times\](x, y, z, t) = a – ib. The product of these two indicates the probability density of finding a particle in space at a time. However, 𝚿2 is the physical interpretation of wave function as it provides the probability information of locating a particle at allocation in a given time.
Choose the Appropriate Option
What did Schrodinger’s equation describe?
Motion of Light
Splitting atom’s process
Wave function complement
Matter waves behaviour
Which of these describes probability density the best?
Wave function’s absolute value
Wave function’s absolute square
Wave function’s square root
Wave function’s inverse
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FAQs on Wave Function
1. What is an acceptable wave function?
An acceptable wave function has to be single-valued, must be normalised, and have to be continuous in the time interval given.
2. What is a normalisation of the wave function?
It is the situation wherein all the probabilities add up to make it 1. In case it adds up to some other constant, then this constant value is incorporated in the wave function to make the probability 1 and attain the state of normalisation.
3. What is the physical significance of wave function?
The wave function's physical significance is none for a particle as it is a complex and non-observable quantity. However, the positive square root of the wave function has physical importance.
4. Describe wave behaviour.
Waves display several basic phenomena. Here are main ones:
Reflection
In reflection, a wave encounters an obstacle and is reflected back. When waves hit a boundary and are reflected, the angle of incidence equals the angle of reflection.
Refraction
In refraction, a wave bends when it enters a medium through which it has a different speed. The speed of a wave depends on the properties of the medium through which it travels. When a wave enters at an angle a medium through which its speed would be slower, the wave is bent toward the perpendicular. And vice versa.
Diffraction
In diffraction, waves bend when they pass around small obstacles and spread out when they pass through small openings.
Interference
In interference, when two waves meet, they can interfere constructively, creating a wave with larger amplitude than the original waves, or destructively, creating a wave with a smaller amplitude or even zero amplitude. The waves from two or more centres of disturbance may reinforce each other in some directions and cancel in others.
5. What is Quantum Mechanics?
Quantum mechanics is the branch of physics relating to the very small. In quantum mechanics, objects instead exist in a haze of probability. The objects have a certain chance of being at point A, another chance of being at point B and soon.
Here are the three revolutionary principles of quantum mechanics.
Quantized properties: Certain properties, such as position, speed and colour, can sometimes only occur in specific amounts. This challenged a fundamental assumption of classical mechanics, which said that such properties should exist only on a smooth, continuous spectrum. Scientists coined the word quantized to best describe the idea that some properties clicked like a dial.
Particles of light: Light can sometimes behave as a particle. This was initially met with harsh criticism, as it ran contrary to 200 years of experiments showing that light behaved as a wave.
Waves of matter: Matter can also behave as a wave. This ran counter to the roughly 30 years of experiments showing that matter, such as electrons, exist as particles.
Quantum mechanics began as a set of controversial mathematical explanations of experiments that the math of classical mechanics could not explain. It began at the beginning of the 19th century around the same time that Albert Einstein published his theory of relativity.
Unlike relativity though, the origins of quantum mechanics cannot be attributed to any one scientist. Multiple scientists contributed to the foundation of the three revolutionary principles that slowly and steadily gained acceptance through experimental verification.
6. What is the history of quantum mechanics?
The term quantum mechanics was coined by a group of physicists that included Max Born, Werner Heisenberg, and Wolfgang Pauli, at the University of Göttingen in the early 1920s. It was first used by Max Born in his 1924 paper titled Zur Quantenmechanik. In the years to follow, this theoretical basis slowly began to be applied to chemical structure, reactivity, and bonding.
In 1927, Heisenberg formulated an early version of the uncertainty principle, analyzing a thought experiment where one attempts to measure an electron's position and momentum simultaneously. But he did not give precise mathematical definitions of what the uncertainty meant. And in the same year, Paul Dirac began the process of unifying quantum mechanics with special relativity by proposing the Dirac equation for the electron.
But the history of Quantum mechanics actually interlaces with the history of quantum chemistry, and began essentially with a number of different scientific discoveries, including the discovery of cathode rays by Michael Faraday (1838), the black-body radiation problem by Gustav Kirchhoff (1859–60 ), the suggestion by Ludwig Boltzmann that the energy states of a physical system could be discrete (1877), the quantum hypothesis by Max Planck (1900).
Further, in 1905, Albert Einstein, in his attempt to explain the photoelectric effect, (deriving from Heinrich Hertz work), postulated consistently with Max Planck's quantum hypothesis that light itself is made of individual quantum particles.