Wave Function Graph
Karriem Drewery
What does the wave function ψ represent? ","acceptedAnswer":"@type":"Answer","text":"The wave function represents the probability of measuring a system to be in some state."},"@type":"Question","name":"What is a wave function?","acceptedAnswer":"@type":"Answer","text":"A wave function in quantum mechanics is a mathematical function that describes the state of a quantum system. Its output is a complex probability-amplitude describing the state of the system, whose modulus squared is the probability of the system to be in that state when measured.","@type":"Question","name":"Why does a wave-function have to be normalized?","acceptedAnswer":"@type":"Answer","text":"As the modulus square of the wave-function is a probability, it must only take values between 0 and 1, else it would violate the laws of probabilities. In order to ensure this holds, the wave function must be multiplied by an overall normalization factor.","@type":"Question","name":"What is wave function with example?","acceptedAnswer":"@type":"Answer","text":"A wave function in quantum mechanics is a mathematical function that describes the state of a quantum system. For example the wave function of a free moving particle is described using a Gaussian wave function. "]} #ab-fullscreen-popup display: none; Find study contentLearning Materials
If two wave functions with amplitudes \(\Psi_1(x)=A\) and \(\Psi_2(x)=B\) interfere with each other, the probability associated with the combined wave function is always given by \(\Psi_12=A^2+B^2\).
Quantum mechanics defines the state of a system probabilistically. This means that one cannot know the precise state of a system before making a measurement. Mathematically, a quantum wave function denoted by \(\Psi\) encodes these probabilities. This quantum wave function is a function of the degrees of freedom defining the possible states of the system. The wave function then outputs a complex number, known as the probability amplitude, whose modulus squared gives the probability density of the system being in that particular state.
Complex numbers are numbers with both a real and an imaginary component. They are of the form \(x+yi\), with \(i\) defined by \(i^2=-1\). The modulus squared of a complex number, \(z=x+iy\), is defined as \[z^2=zz^*=(x+iy)(x-iy)=x^2+y^2.\]
The wave function contains all the information about a system and how it evolves over time. However, the laws of quantum mechanics restrict our experimental access to this information. Born's rule, one of the fundamental postulates of quantum mechanics, describes the relationship between the wave function and the probability density associated with the likelihood of measuring the system to be in some state.
The above inequality, which appears across many fields of research, is called the triangle inequality. Any time we want to establish a distance measure in an arbitrary geometrical space, we have to appeal to it. Because it's so general, mathematicians regard the triangle inequality as a fundamental result.
Fig. 1. Image showing the probability densities of an electron around a nucleus in a hydrogen atom. The lighter the color the more likely the electron will be found there. This probability density is defined by the wavefunction \(\Psi\).
The energy level or orbital of the electron within the atom determines the shape of the probability cloud. Figure one shows the orbitals of an electron within a hydrogen atom. The electron is most likely to be found in the lighter regions whilst it will never be found in the black regions. In addition, the so-called quantum numbers characterize the shape of the orbital by defining the angular momentum and spin of the electron. At low energies, the orbital has a ring-like shape similar to the Bohr Model of circular orbits. However, at higher energies, the shape of these orbitals becomes more complex. This wave function model for atomic electrons is the most accurate picture of the atom we have and has been key to developing our theory of chemical bonding. Consequently, one of quantum mechanics' most important applications is its prediction of the properties of the periodic table.
We can visualize the behavior of a quantum particle with the graph of its wave function. By looking at the graph's amplitude, we can see which regions are most likely to contain the particle. Charting how the graph changes over time also indicates how the quantum particle evolves. For example, look at the graph below representing the wave function of a particle in a box. The wave function is a standing wave with four nodes and three anti-nodes. The nodes of the particle's wave function are the regions where the probability of finding the particle are lowest, whereas the anti-nodes are regions where the probability to find the particle is highest.
Remember it is \(\Psi(x)^2\) that gives the probability of measuring the particle at the position \(x\). Negative values of the wave function do not represent low probabilities, they simply represent a phase difference.
Fig.2- A graph of the wave function of a particle trapped in a box. The standing wave formed shows that there are zones of high probability and areas of zero probability, much like the electron orbitals in an atom.
It is important to keep in mind that the wave function of a system determines the probability of finding the system in some state or position upon measurement. However, something strange happens once we do make that measurement. Let's say we wanted to check the results of our first measurement by making a second measurement shortly after. For the result of our first measurement to be valid, this second measurement must return the same result. Otherwise, there is nothing to confirm our initial measurement. This means that once measured, a quantum system must remain in the same state. This has a profound implication for our understanding of the wave function.
Consider measuring the position of a particle, again described by some wave function. Initially, the wave function is a spread of probability-amplitudes over space. However, once we make a measurement we know for certain the position of the particle. Let's call this position \(C\). Any further measurement must always return \(C\), thereby making the probability at \(C\) one and zero everywhere else. This means the wave function becomes a sharp peak centered at \(C\) with zero amplitude everywhere else.
We say that the measurement has 'collapsed' the wave function down to a single point. This collapse of the wave function demonstrates the mysterious nature of measurements in quantum mechanics. It is one of the strangest and most controversial aspects of quantum physics. The physics behind such a collapse, and whether such a physical interpretation is even possible, is still a topic of fierce debate.
Asking what happens during wave function collapse is such a difficult conceptual problem that many physicists have given up on trying to solve it altogether. Instead, they adopt a "shut up and calculate" mentality in which they trust the ability of the mathematics of quantum mechanics to make successful predictions without worrying too deeply about the meaning of the theory. Philosophers of physics, on the other hand, do care significantly about establishing the meaning of quantum mechanics. By studying the historical development and logical foundations of quantum mechanics, they hope to solve this enigma once and for all.
A wave function in quantum mechanics is a mathematical function that describes the state of a quantum system. Its output is a complex probability-amplitude describing the state of the system, whose modulus squared is the probability of the system to be in that state when measured.
As the modulus square of the wave-function is a probability, it must only take values between 0 and 1, else it would violate the laws of probabilities. In order to ensure this holds, the wave function must be multiplied by an overall normalization factor.
A wave function in quantum mechanics is a mathematical function that describes the state of a quantum system. For example the wave function of a free moving particle is described using a Gaussian wave function.
In quantum physics, a wave function (or wavefunction) is a mathematical description of the quantum state of an isolated quantum system. The most common symbols for a wave function are the Greek letters ψ and Ψ (lower-case and capital psi, respectively). Wave functions are complex-valued. For example, a wave function might assign a complex number to each point in a region of space. The Born rule[1][2][3] provides the means to turn these complex probability amplitudes into actual probabilities. In one common form, it says that the squared modulus of a wave function that depends upon position is the probability density of measuring a particle as being at a given place. The integral of a wavefunction's squared modulus over all the system's degrees of freedom must be equal to 1, a condition called normalization. Since the wave function is complex-valued, only its relative phase and relative magnitude can be measured; its value does not, in isolation, tell anything about the magnitudes or directions of measurable observables. One has to apply quantum operators, whose eigenvalues correspond to sets of possible results of measurements, to the wave function ψ and calculate the statistical distributions for measurable quantities.
In the 1920s and 1930s, quantum mechanics was developed using calculus and linear algebra. Those who used the techniques of calculus included Louis de Broglie, Erwin Schrdinger, and others, developing "wave mechanics". Those who applied the methods of linear algebra included Werner Heisenberg, Max Born, and others, developing "matrix mechanics". Schrdinger subsequently showed that the two approaches were equivalent.[15]
c80f0f1006