What is Wave function: Definition and 873 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.
Hi all, I've got a question I think I understand conceptually but not mathematically...
Homework Statement
A transverse sinusoidal wave on a string has a period T = 25.0 ms and travels in a negative x direction with a speed of 30.0 m/s.
At t=0, a particle on the string at x=0 has a transverse...
I want to know whether the wave function of particle is unique? If not, could we find a ψ to rationalize the equation Pψ=Aψ, in which P is the momentum operator and A is a constant. Thank you!
XXZ model in 1-dimension, with external field, what is the exact form of its ground state wave function? Recently I've read the ground state wave function of XY model and was curious about the condition of XXZ. Do they have a similar manner?
The XY result is attached below.
Thanks.
Homework Statement
Hi all, I've been given some homework on wave functions to find direction and velocity etc... but as its the first week of uni for me, we haven't even covered wave functions or anything :/ and the homework is due on weds 10th oct...
Homework Equations...
I've heard some people say that the wave function and its first derivative must be continuous because the probability to find the particle in the neighborhood of a point must be well defined; other people say that it's because it's the only way for the wave function to be physically significant...
Homework Statement
Just a snipit of one of my homework problems. I'm trying to find out what \Psi \frac{\partial \Psi^{*}}{\partial x} equals to help me find out what the probability current for a given free particle is.
Homework Equations
\Psi = Ae^{i(kx-\frac{\hbar k^{2}t}{2m})}...
Why is it that when observing an electron or photon it causes the wave function to collapse, surely the photons that cause this collapse were still "colliding" with it when we wern't looking. Or does it only collapse the wave function from the observers viewpoint?
(I'm abit of a noob so if...
Uncertainty - Harmonic Oscillator
The Wave function for the ground state of a quantum harmonic oscillator is
\psi=(\alpha/\pi)^{1/4}e^{-\alpha x^2/2}
where \alpha = \sqrt{ mk/ \hbar^2} .
Compute \Delta x \Delta p known:
Heisenberg Uncertainty Principle:
\Delta p \Delta x >= \hbar/2...
Hi...I am new to this forum.
Can somebody clear a fundamental doubt i have?:uhh: A wave function has a form found by applying Schrodinger's equation. In steady-state systems, arent the system eigen functions, the wave equation of the system? if so is it the energy eigen function or the momentum...
Hi,
I'm looking at this wave function:
\psi(x,t) = \frac{4}{5}{\psi}_{1} + \frac{3}{5}{\psi}_{2}
The functions involved here are the typical eigenfunctions for the ground state and first excited level in an infinitely-deep 1-D square well.
Defining
A = 4/5.\sqrt{2/a}
B =...
Is it correct to think, that with a scalar complex Klein-Gordon field the wave function \Psi:\mathbb{R}^3\to\mathbb{C} of one particle QM is replaced with an analogous wave functional \Psi:\mathbb{C}^{\mathbb{R}^3}\to\mathbb{C}? Most of the introduction to the QFT don't explain anything like...
A wave function (psi) equals A(exp(ix)+exp(-ix) in the region -pi<x<pi and zero elsewhere.
Normalize the wave function and find the probability of the particle being between x=0 and pi/8
Equation is : the integral of psi*(x,t)psi(x,t)=1 for normalization
Homework Statement
I have a problem in which I have a two-atomic molecule, and I'm supposed to find the energy and wave function in the ground state, given the particles' masses m_1,m_2 and the potential V(r)=kr^2, where r is the distance between the particles.
I don't necessarily need this...
Homework Statement
phi(r,0)=1/rt(2)(phi1+ph2)
What is phi(r,t)?Homework Equations
The Attempt at a Solution
Is this simply a case of introducing a phase constant to the eqn. So:
phi(r,t)=1/rt(2)(phi1+phi2)e^i(theta)t
or do we need to modify phi1 and phi2.
Hi!
How can I write the wave function of a particle in an infinite box (in the state n) as a superposition of the eigenstates of the momentum operator?
the wave function is:
PHIn(x,t) = sqrt(2/a) * Sin(n * PI/a * x) * exp(-i En/h * t)
Thanks for your help!
If I have af wavefunction that is a product of many particle wavefunctions
$\Psi = \psi_1(r_1)\psi_2(r_2) ... \psi_n(r_n)$
If I then know that the parity of $ \Psi $ is even. Can I then show that the wavefunction i symmetric under switching any two particles with each other. That is...
Homework Statement
Use the ground-state wave function of the simple harmonic oscillator to find: Xav, (X^2)av and deltaX. Use the normalization constant A= (m*omegao/(h_bar*pi))^1/4.
Homework Equations
deltaX=sqrt((X^2)av-(Xav)^2)
wavefunc=A*e^(-ax^2) ?
The Attempt at a...
Hello
I know that every particle has attached a wavefunction in quantum mechanics.
How does a free particle move in quantum mechanics?
The wavefunction has periodic zeroes, and the |wavefunction|^2 gives the probability of finding that particle...so does this mean that in space the...
I have a couple questions about finite potential barriers that I can't seem to figure out on my own...
1) Why does the real part of the wave function collapse inside the barrier (square, rectangular, barrier with V less than the energy of particle)? It seems to me that there should be some...
Homework Statement
While solving the integral of a wave function,I came across the term cos(n*pi) , where n is an integer. Is that term equal to +1 or -1 (I know that it could be either one depending on whether n is odd or even) but how do I proceed with the integral?
Homework...
"(a) Use the radial wave function for the 3p orbital of a hydrogen atom (see Table 15.2) to calculate the value of r for which a node exists.
(b) Find the values of r for which nodes exist for the 3s wave function of the hydrogen atom."
For part a, I looked at Table 15.2 and found the equation...
I am just learning QED and could not understand the role of wave function. Is the basic equation in QED the Schrodinger Equation? Is the difference between Quantum mechanics and QED just they have different Hamiltonians.
I have tried to read the original paper of Tomonaga in 1946 Progress of...
what would a wave function of
phi(x) = Ke^(-a|x|)
look like?
would it be like an exponential graph with a graph reflected along the y axis?
and its probability distrubution (phi)^2?
i have no idea...i can't seem to find it by googling.
Hi all,
I've got a tough problem that I need some guidance on.
Question: Consider a wave function that is a combination of two different infinite-well states, the nth and the mth...
we often say that the square of wave function gives us the probability density where the particle is. how can the square of a function might predict about the existence of a particle?
Question from textbook (Modern Physics, Thornton and Rex, question 54 Chapter 5):
"Write down the normalized wave functions for the first three energy levels of a particle of mass m in a one dimensional box of width L. Assume there are equal probabilities of being in each state."
I know how...
When doing the initial conditions of the velocity of the wave function, why do they have a position (x) derivative (i.e. cF'(x)-cG'(x)=h(x)).
It appears in here.
http://en.wikipedia.org/wiki/D%27Alembert%27s_formula
How someone explain how the c came about and why position derivatives are...
I found a layman's explanation of the wave characteristics of subatomic particles in the form of a "Dr.Quantum" video from "What the Bleep do we know?". Aside from the parapsychological junk in the last 2/3rds of the movie, the explanations of quantum properties seemed mostly accurate and...
A wave function psi = 3i|up> + 1|down> corresponding to the spin of the electron.
If I want to draw the distribution of the measured outcome, do I do the following?Probability of spin up = 0.9
Probability of spin down = 0.1
So I would draw a bar graph showing that spin up has a value of 0.9...
I solved the differential equation for theta portion of the hydrogen wave function using a power series solution. I got a sub n+2 = a sub n ((n(n+1)-C)/(n+2)(n+1)). I then truncated the power series at n = l to get
C= l(l+1).
I know need to use the recursion formula I found to find the l =...
how fast does the wave function "decollapse"?
as you know the wave function (indeterminacy) may collapse due to a measurement. However, after a measurement it returns after a while to its initial state of indetermincy. (I don't know what to call this transition back to indetermincy; is there a...
The following is a "ooze" wafve function:
\Psi_{ooze} (x,t)=\frac{1}{K} \left( \Psi_1 + \Psi_2+...+\Psi_{1000} \right)
1. I am to find the value of K, but I don't even know what it represents. Is K the coefficent to normalize the probablity to 1?
2. Probability where energy E_q...
I'm having problem with griffith QM problem 4.43:
Construct the spatial wave function for hydrogen in the state n = 3, l =2, m = 1. Express your answer as a function of r, \theta, \phi, and a (the Bohr radius) only.
My prof. gave hints about radial wave function, but I have no idea how to...
the wave function descrbing a state of an electron confined to move along the xaxis is given at time zero by:
\Psi(x,0)=Ae^{\frac{-x^2}{4 \sigma^2}}
where sigma is a constant (i believe).
I am asked to find the probability of finding the electron in a degion dx centered at x=0.
I...
My intuition is that it would be unitless. But if it's magnitude squared is a probability density, then its units would have to be 1 over some power of length. Specifically 1/L^(n/2) where n is the dimension. Where's the error in my thought? Thanks
Hello. In QM we can determine the probability of any event ocurring given the wavefunction. Once we actually take a measurement the particle 'picks' a state to be found in.
so my question is how do we know a priori that the particle is in two or more states at the same time before we make a...
how do you find the normalized wave functions of the hydrogen atom for n=1, l=0 and ml=0?
in my textbook, it's a table, but i have no idea where the figures come from...
why does the wave function have to be normalizable, and why does it have to go to 0 and x approaches positive/negative infinity and y approaches positive/negative infinity ?
Scientists wan't to know where wave function collapse occures. I have found at least one.
Take the experiment of shooting electrons through two holes that are close together and seeing where they land at a detection screen on the other side. If you shut one hole you get a particle or lump...
I need to explain why, as the energy of a bound state in a finite potential well increases, the wave function extends more outside of the well. I need to do this from both a mathematical and a physical point of view. I think I know the mathematical explanation (see attached image). Can anyone...
for the following question:
the wave function of a moving particle (a) reflects the probablity of finding the particle at a particular place and all the time (b)reflects the probablity of finding the particle at a particular place and a particular time (c)reflects the probablity of finding the...
I didn't find an article about this so I'll ask it here:
How does the quantum mechanical wave function of a particle arise in ST?
Sorry if it has been asked before, but I couldn't find a topic about it.
Normalization of a wavefunction
Let Phi be a wave function,
Phi(x)= Integral of {exp(ikx) dk} going k from k1 to k2
I'm having trouble normalizing the wave function. I calculated the integral, then multiply by its conjugate and now I'm supposed to integrate again /Phi(x)/^2 in all...