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 a wave function in quantum mechanics, and what does it represent?
In quantum mechanics, a wave function, represented by the Greek symbol Psi (Ψ), is a mathematical function that provides a complete description of the quantum state of a particle or system. It does not describe a physical wave like a water wave, but rather an abstract probability amplitude. The function itself contains all the measurable information about the particle, such as its position, momentum, and energy, as a function of time and space.
2. What is the key difference between the wave function (Ψ) and its square (∣Ψ∣²)? Which one has physical significance?
The primary difference lies in their physical interpretation. The wave function (Ψ) is often a complex number and has no direct physical meaning on its own. However, the square of its absolute value, ∣Ψ∣², has a crucial physical significance. It represents the probability density of finding the particle at a specific point in space at a given time. While Ψ is a mathematical tool, ∣Ψ∣² is the value that connects the theory to measurable reality.
3. What are the essential properties of an acceptable or 'well-behaved' wave function?
For a wave function to be physically realistic as per the CBSE/NCERT curriculum, it must satisfy several conditions:
- Finite: The wave function must have a finite value everywhere. An infinite value would imply an infinite probability of finding the particle, which is not possible.
- Single-valued: At any given point in space and time, the wave function must have only one value to ensure there is a single, unique probability for the particle's location.
- Continuous: The wave function and its first derivative must be continuous. This ensures that the probability density is well-defined and does not have abrupt, physically impossible jumps.
4. How is a quantum wave function different from a classical wave, such as a sound or light wave?
A quantum wave function and a classical wave are fundamentally different concepts. A classical wave, like sound, is a disturbance that propagates through a physical medium, carrying energy. A quantum wave function (Ψ), on the other hand, is not a physical entity. It is a mathematical construct that exists in an abstract space and describes the probability amplitude of a particle's quantum state. It doesn't carry energy in the same way; instead, its square gives the probability of locating the particle.
5. How is the specific wave function for a particle determined?
The wave function for a particle in a given physical situation is not a universal formula. Instead, it is found by solving the Schrödinger equation for that specific system. The equation takes into account the particle's mass and the forces acting upon it (represented by the potential energy function). The solution, Ψ, is unique to the system's boundary conditions and energy, providing the precise wave function for that scenario.
6. What does it mean for a wave function to be 'normalised', and why is this concept important?
Normalisation is the process of scaling a wave function so that the total probability of finding the particle somewhere in all of space is exactly 1 (or 100%). This is a fundamental requirement because the particle must exist somewhere. The condition for normalisation is that the integral of ∣Ψ∣² over all space equals 1. This ensures that the probabilistic interpretation of the wave function is consistent and physically meaningful.
7. Can you provide a simple example of what a wave function describes for a particle?
A classic example is a particle in a one-dimensional box. The wave function for this particle is zero outside the box, meaning there is a 0% chance of finding it there. Inside the box, the wave function takes the form of a sine wave. The square of this sine wave, ∣Ψ∣², shows that the particle is most likely to be found at the peaks (antinodes) of the sine wave and will never be found at the points where the wave is zero (nodes), except at the boundaries. This illustrates how the wave function governs the probability of the particle's location.
