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 complexvalued 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 (lowercase 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., zcomponent of spin) – these values are often displayed in a column matrix (e.g., a 2 × 1 column vector for a nonrelativistic 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 nonrelativistic quantum mechanics,
the squared modulus of the wave function, ψ2, 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.
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...
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...
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...
Hello, I wonder if it is possible to write Bloch wave functions in momentum space.
To be more specific, it would calculate something like (using Sakurai's notation):
$$ \phi(\vec k) = \langle \vec k  \alpha \rangle$$
Moving forward in a few steps:
Expanding:
$$ \phi(\vec k) = \int d^3\vec r...
I would like to know if any of you think there's any sort of connection, analogy, or common features between, sets in set theory and wave functions in QT?
Wave functions lack trajectories, so do sets. Wave functions also distribute over areas, as sets can do. To my understanding, wave...
I saw this statement from the textbook "Quantum physics of atoms, molecules, solids, nuclei, and particles" second edition pg 166. According to the text, is the author saying the solution to the TISE is the eigenfunction and when you multiply the time dependent part, you get the wave function? I...
In classical mechanics, we have either Newton’s laws or a Lagrangian in terms of coordinates and their derivatives (or momenta) and we can solve them for the behavior of the system in terms of these variables, which are what we observe (measure).
In QM, we quantize classical mechanics by making...
Hello, I was going to solve with a calculator the eigenvalues problem of the Schrödinger equation with Yukawa potential and I was thinking that the boundary conditions on the eigenfunctions could be the same as in the case of Coulomb potential because for r > 0 the exponential term goes to 1...
Hello,
I would like to write a product of two wave functions with a single ket. Although it looks simple, I do not remember seeing this in any textbook on quantum mechanics. Assume we have the following:
##\chi(x) = \psi(x)\phi(x) = \langle x  \psi \rangle \langle x  \phi \rangle##
I would...
Is the wave function for the positron the complex conjugate of the wave function for the electron? I've tried to google this, but I can't seem to get a definite answer from a reliable source. It seems that antimatter is derived in quantum field theory which does not concentrate on wave...
The question is as follows:
A particle of mass m has the wave function
psi(x, t) = A * e^( a ( ( m*x^2 / hbar) +i*t ) )
where A and a are positive real constants.
i don't know how to format my stuff on this website, so it may be a bit harder to read. Generally when i write "int" i mean the...
Suppose we have a particle, let's say an electron, in a box of size D. And we add another one next to it at some distance L center to center. How do we solve for the wavefunctions of the electron. Can it be solved in normal QM or do we need QFT. Thanks.
1. Homework Statement
I've been using a recurrence relation from "Adv. in Physics"1966 Nr.57 Vol 15 . The relation is :
where Rnl are radial harmonic oscillator wave functions of form:
The problem is that I can't prove the relation above with the form of Rnl given by the author(above). I've...
Hi,
Apologies if this questions is really easy but it is something quite subtle which is annoying me. In my book of quantum physics it gives a wave function of definite momentum:
ψ = Aeipx/ħ
It goes on to say that since there is a momentum 'p' in the exponential then the momentum is known...
How come a+a ψn = nψn ? This is eq. 2.65 of Griffith, Introduction to Quantum Mechanics, 2e. I followed the previous operation from the following analysis but I cannot get anywhere with this statement. Kindly help me with it. Thank you for your time.
Do higher dimensional branes, like the super membrane (which is a 2D brane) or the NS5/M5 brane, have wave functions? I know that they become unstable once they are quantized, but does that mean that they do not have wave functions? You will never here about any thing regarding an M2 wave...
For example, I am following the below proof:
Although the above derivation involves a projection on the position basis, it appears one can generalize this result by using any complete basis. So despite it not being explicitly mentioned here, are all wave functions with any continuum basis...
Hello all,
First of all, I am aware that dissonance and consonance between pitches also depend to an extent by culture and musical origin but there also seems to be some degree of objective perception among people that can be explained scientifically. Also, I'm very new to this so I could be...
Hi!
For the probability interpretation of wave functions to work, the latter have to be square integrable and therefore, they vanish at infinity. I'm reading Gasiorowicz's Quantum Physics and, as you can see in the attached image of the page, he works his way to find the momentum operator. My...
Homework Statement
Is there a physical difference between the following wave functions? If yes, why? If no, why not?
\Psi(x,0) =5e^{ax^2}
\Psi(x,0) =\frac{1+i}{\sqrt{3}}e^{ax^2}
\Psi(x,0) =e^{i\pi/7}e^{ax^2}
Homework Equations

The Attempt at a Solution
They only differ in the...
Homework Statement
I am unsure if the first statement below is true.
Homework Equations
\frac{\partial \psi^*}{\partial x} \frac{\partial^2 \psi}{\partial x^2}=\frac{\partial^2 \psi}{\partial x^2}\frac{\partial \psi^*}{\partial x} Assuming this was true, I showed that \int \frac{\partial...
Homework Statement
Assume a particle with a wave function ##\psi(x)## such that ##\infty < x < \infty##, that move under some potential ##V(x)##.
Show that:
a) two wave functions with same energies can only differ by a complex phase;
b) if the potential is real, then you can choose the wave...
Why is the triplet state space wave function ΨT1=[1σ*(r1)1σ(r2)1σ(r1)1σ*(r2)] (ie. subtractive)? How does it relate to its antisymmetric nature?
Also, why is this opposite for the spin wave function α(1)β(2)+β(1)α(2) (ie. additive)? And why is this one symmetric even though it describes the...
I have read recently that the motion of an electron of momentum p must be described by the means of a plane waves :\psi(\vec r,t)=Ae^{i(\vec k \cdot \vec r wt)}=Ae^{i(\vec p\cdot \vec r Et)/\hbar}
de Broglie hypothesis states that every particle of momentum p has a wavelength lamda.
I will...
What's the type of wave functions? Is it just: function from a point in spacetime to Z; takes a location and returns an amplitude in discrete units?
(bonus question: according to your favorite theory, what is the type of points in spacetime (that is, topology of spacetime)? is it like r^n for...
Can the general solution to the Schrodinger equation be asymmetric (has neither even or odd solutions)?
Question (1): I saw somewhere that you cannot have a solution that is both it must be either odd or even, and I was wondering: why?
I was working on a problem where the potential function...
Hello,
I need to create a 2D electron energy density plot in Mathematica to compare with my STM experimental results in my lab class. This would be done by plotting the superposition of the symmetric and antisymmetric wave functions,
$$\Psi_s(\textbf{r}) =...
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
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...
Homework Statement
Homework EquationsThe Attempt at a Solution
For x>b, Ψ(x) = Aeikx + Beikx , where k = (√2mE)/hbar
a<x<b Ψ(x) = Ceik'x + Deik'x , where k = (√2m(U2  E)/hbar
This is the problem part
0<x<a Ψ(x) = Fsink''x...
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...
I am working on a PDE problem like this:
Consider the wave equation with homogeneous NeumannDirichlet 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<...
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(xct)) + 1/2c ∫ g(z) dz
The...
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...
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...
Homework Statement
Homework EquationsThe Attempt at a Solution
http://i.imgur.com/tktQBsp.jpg
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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...
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...
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 2D or 3D graph (and I can't...
Homework Statement
The ground, 1st excited state and 2nd excited state wave functions for a vibrating molecules are:
Ψ0(x) = a eα2/2
Ψ1(x) = b (x+d) eα2/2
Ψ2(x) = c (x2 + ex + f) eα2/2
respectively, with constants: a, b, c, d, e and f. Use the fact that these wave functions are part of an...
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:
<UV> = Det{<uivi>}
Thank you
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...
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"...
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...
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...
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)=ex2/2 and ψ1(x)=xex2/2 are eigenfunction of H with eigenvalues ½ and 3/2, respectively.
Find the value of the coefficient a such that...
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...