# What is Wave functions: Definition and 153 Discussions

A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum 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. The most common symbols for a wave function are the Greek letters ψ and Ψ (lower-case and capital psi, respectively).
The wave function is a function of the degrees of freedom corresponding to some maximal set of commuting observables. Once such a representation is chosen, the wave function can be derived from the quantum state.
For a given system, the choice of which commuting degrees of freedom to use is not unique, and correspondingly the domain of the wave function is also not unique. For instance, it may be taken to be a function of all the position coordinates of the particles over position space, or the momenta of all the particles over momentum space; the two are related by a Fourier transform. Some particles, like electrons and photons, have nonzero spin, and the wave function for such particles includes spin as an intrinsic, discrete degree of freedom; other discrete variables can also be included, such as isospin. When a system has internal degrees of freedom, the wave function at each point in the continuous degrees of freedom (e.g., a point in space) assigns a complex number for each possible value of the discrete degrees of freedom (e.g., z-component of spin) – these values are often displayed in a column matrix (e.g., a 2 × 1 column vector for a non-relativistic electron with spin 1⁄2).
According to the superposition principle of quantum mechanics, wave functions can be added together and multiplied by complex numbers to form new wave functions and form a Hilbert space. The inner product between two wave functions is a measure of the overlap between the corresponding physical states, and is used in the foundational probabilistic interpretation of quantum mechanics, the Born rule, relating transition probabilities to inner products. The Schrödinger equation determines how wave functions evolve over time, and a wave function behaves qualitatively like other waves, such as water waves or waves on a string, because the Schrödinger equation is mathematically a type of wave equation. This explains the name "wave function", and gives rise to wave–particle duality. However, the wave function in quantum mechanics describes a kind of physical phenomenon, still open to different interpretations, which fundamentally differs from that of classic mechanical waves.In Born's statistical interpretation in non-relativistic quantum mechanics,
the squared modulus of the wave function, |ψ|2, is a real number interpreted as the probability density of measuring a particle as being at a given place – or having a given momentum – at a given time, and possibly having definite values for discrete degrees of freedom. The integral of this quantity, over all the system's degrees of freedom, must be 1 in accordance with the probability interpretation. This general requirement that a wave function must satisfy is called the normalization condition. 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.

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1. ### A Can particles appear and disappear "with" a cause?

The first thing we need for this is to define what a particle is... It is an object that has specific intrinsic properties and is described by a wave sign How to measure it? This is done by the interaction of the particle to be measured with the measurement system. When measuring, the wave...
2. ### A LCAO graphene orbitals wave functions

Hello, My name is Josip Jakovac, i am a student of the theoretical solid state physics phd studies. First I want to apologize if my question has already been answered somewhere here, I googled around a lot, and found nothing similar. My problem is that I need to apply TBA to Graphene. I went...
3. ### B What proof do we have of wave functions?

How can we be sure that a system on the scale of atoms can be described by a single scalar field or the wave function ##\psi##. I don't just want to do shut up and calculate, maybe using a wave function and then putting it through the time evolution of the Schrödinger equation works, but why...

30. ### Tight Binding Wave Functions

Dear all, Could somebody please, indicate me some tutorial, in order to generate a 3D grid to plot the wave function using the Hamiltonian eigenvalues and the slater type orbitals ? Thanks in advance, Wellery
31. ### Do wave functions go to zero at ~ct?

Suppose you have a free election and you make a measurement of its position r_0 at time t = 0. You then wait some time t required for the wave function to evolve out of its collapsed eigenstate. The electron now supposedly has a wave function expanding to infinity in all directions, albeit with...
32. ### Where does band index come from in block wave functions?

Can you show me as explicitly as possible through equations?
33. ### Potential Step and Wave Functions

Homework Statement Homework EquationsThe Attempt at a Solution For x>b, Ψ(x) = Ae-ikx + Beikx , where k = (√2mE)/hbar a<x<b Ψ(x) = Ce-ik'x + Deik'x , where k = (√2m(U2 - E)/hbar This is the problem part 0<x<a Ψ(x) = Fsink''x...
34. ### Wave function with a certain wavelength

I have a number of questions about the wave function - 1. Do photons have wave functions like the one in Schrodinger equation? 2. If they do, when you send out a wave function with a certain wavelength, then because you know the momentum with no uncertainty the uncertainty of the position...
35. ### Physical interpretation of Neumann-Dirichlet conditions

I am working on a PDE problem like this: Consider the wave equation with homogeneous Neumann-Dirichlet boundary conditions: ##\begin{align} u_{tt} &= c^2U_{xx}, &&0<x<\mathscr l, t > 0\\ u_x(0, t) &=u(\mathscr l, t) = 0, &&t > 0\\ u(x, 0) &=f(x), &&0<x< \mathscr l\\ u_t(x, 0) &=g(x), &&0<x<...
36. ### Solve Wave Equation: e^(-x^2), x*e^(-x^2), -infinity<x<infinity

Homework Statement So it says solve this wave equation : [y][/tt] - 4 [y][/xx] = 0 on the domain -infinity<x<infinity with initial conditions y(x,0) = e^(-x^2), yt(x,0) = x*(e^(-x^2)) Homework Equations I used the D Alembert's solution which is 1/2(f(x+ct)+f(x-ct)) + 1/2c ∫ g(z) dz The...
37. ### Systems with more than 1 wave function

Hello,I am new to quantum mechanics.I just want to clear this equation: ψ(x) = ∑n anψn(x) What does this actually mean?Is this equation telling us that the system is moving as a wave? Or,as I think,for example let's suppouse we have 2 electrons in a system,and the wave function becomes this...
38. ### Normalizing Wave Functions

Hi, so I'm having a bit of trouble understanding the normalization of radial waves. I understand that the equation is the integral of ((R^2)r^2)=1 but I'm not understanding how the process works. I need the normalization constant on R32. I got the function to come out to be...
39. ### Orthogonality of wave functions

Homework Statement Homework EquationsThe Attempt at a Solution http://i.imgur.com/tktQBsp.jpg [/B] I assume that you need to prove that the integral of psi1*psi0 is 0, so I have written out the integral and attempted to solve using integration by parts, but whichever way I write out the...
40. ### Normalizing wave functions / superposition

A remote control shot a single photon at a window that has a 50% chance of transmitting and 50% chance of reflecting photons. Once the photon wavepacket hits the glass (without an observer present), it propagates as a superposition of states, one in which it was transmitted and one in which it...
41. ### Help graphing wave functions and probability densities.

Homework Statement Homework Equations The Attempt at a Solution Well, I felt like the first part wasn't too bad and graphed the potential like so: However, to be honest, I'm not even sure if I did that right, as I wasn't sure whether he wanted it as a 2-D or 3-D graph (and I can't...
42. ### Finding constants for 3 wave functions

Homework Statement The ground, 1st excited state and 2nd excited state wave functions for a vibrating molecules are: Ψ0(x) = a e-α2/2 Ψ1(x) = b (x+d) e-α2/2 Ψ2(x) = c (x2 + ex + f) e-α2/2 respectively, with constants: a, b, c, d, e and f. Use the fact that these wave functions are part of an...
43. ### <U|V> overlap integral of two many-electron determinant wave functions

Hello, If we let U and V be two single determinant wave functions built up of spin orbitlas ui and vj respectively, will the overlap between them be as follows: <U|V> = Det{<ui|vi>} Thank you
44. ### Question about collapsing wave functions

So, from my understanding, some particles like electrons exist as a particle and a wave/probability field. What I was wondering, was that when the wave function collapses, is its location determined on the actual location of the particle, which we just can't measure and so represent it as a...
45. ### Query about quantum superposition and wave functions

Hi Everyone I have four questions about the nature of quantum superposition and wave functions: 1. If a particle is quantumly superpositioned in more than one location then as soon as the slightest evidence of the particle's existence in one of the locations is detected by a "measurement"...
46. ### Are all wave functions energy-eigenstates?

So I was reading this http://www.oberlin.edu/physics/dstyer/TeachQM/misconnzz.pdf, a list of common misconceptions students have after an intro course in QM. I'm aware that energy eigenstates are the wave functions at "time = zero" and thus do not completely describe the system. However, it is...
47. ### Hydrogen atom wave functions

Homework Statement I solved the Schrödinger equation, obtaining a wave function in terms of Radial and the spherical harmonics as follows: $$Ψ(r,0)= AR_{10} Y_{00} + \sqrt{\frac23} R_{21} Y_{10} + \sqrt{\frac23} R_{21} Y_{11} - \sqrt{\frac23} R_{21} Y_{1,-1}$$ Homework Equations...
48. ### Show that the two wave functions are eigenfunction

Homework Statement Consider the dimensionless harmonic oscillator Hamiltonian H=½ P2+½ X2, P=-i d/dx. Show that the two wave functions ψ0(x)=e-x2/2 and ψ1(x)=xe-x2/2 are eigenfunction of H with eigenvalues ½ and 3/2, respectively. Find the value of the coefficient a such that...
49. ### Orthogonality of wave functions to negative momentum states

This is a question I have about the textbook discussion, so I'll do away with the standard format. The author of my QM book (Shankar, Principles of Quantum Mechanics) used the term "negative momentum states," all of a sudden, and I've never heard of it before. He has a little note saying that...
50. ### Can second quantization on strings create states out of the vacuum?

Does anything connect the discrete wave functions? I thought they were suppose to be connected.