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.
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...
Hey!
Here is one that I thought would be easy:
Two traveling waves move on a string that has a fixed end at x=0. They are identical except for opposite velocities. Each has an amplitude of 2.46mm, a period of 3.65ms, and a speed of 111m/s. Write the wave function of the resulting standing...
A short question:
I've learned that the wave function corresponding to a free particle has this form:
Psi(x,0)=1/sqrt(2*Pi)*Integral[g(k)*E^(ikx)dx] (i can't write it in Latex, sorry)
Is it just for the free particle, or any quantum state of a system can be represented in this form...
The Second Ring of Life; The Vesica Attractor
by Christopher Humphrey
Abstract
The fossil record shows a disparity in the formation of complex body plans.
The individual eukaryote cannot build these structures. They do not carry within themselves a blue print for an overall structure...
i am having difficulty with the wave function.
for example in the exercise we are told to write the expression :
a cos x+ b sin x in the form k cos(x-a)
This i had little problem with and was able to work out what quadrant i shoul use etc what i mean is the
all positive. sin positive...
Hey, if anyone can throw in some thoughts I am a little lost. Not sure If I need to integrate, or what. Thanks for any help.
The wave function for a hydrogen atom in the 2 s state is:(attachment)
I need to Calculate the probability that an electron in the 2 s state will be found at a...
What if complex biological systems emerged as a result of a wave function firstly, and the biochemical components followed after. Can this new perspective better explain organizational gaps in evolution.
The Phi-Wave Aether: a Wave Theory of Everything
Caroline H Thompson...
Hey. I am pretty confident i have solve this problem. I just solve the integral of the given wave function, with the given limits... However, I am having a difficult time integrating it. The sqrt(2/L) can be brought outside of the integral, but what can i with the sin function?
The wave...
Hi, I have a question about the mathematical requirements of a wave function in a potential that is infinite at x \leq 0. (At the other side it goes towards infinity at x = \infty.) Now, given a wave function in this potential that is zero for x = 0 and x = \infty. Does it matter what that...
How exactly does one find a wave function? Specifically, I am asked to find the momentum space wave functoin for the nth stationary state in an infinite square well. Then I am to graph the probability density (phi sqaured) for the first and second energy levels. Lastly, I need to use the...
Hi there.
We always put the time dependent part of the wave functions as e^(iwt).
Of course there is a reason! but I don't know it.
Can you help me?
Thanks in advance.
Somy :smile:
On My Last Straw Trying to Find a Wave Function
I am horribly confused as how to I can actually find a wave function for any given problem. The specific wave function I am trying to find right now is that of a neutron passing through a double slit apparatus. Here is how I have the problem set...
hey who can help me with this physics problem?
A particle of mass m is in the state:
Ψ (x, t) = Aexp[-a(sqrt (mx^2) / h)-i (at / sqrt(m )) ]
where A and a are positive real constants.
a) Determine A.
b) What is the frequency ƒ associated with the wave function of this particle?
Explain...
I'm reading an introductionary text on quantum physics and am stumbling a bit with the terms used.
The text discusses a finite potential box (one dimension, time independent). It calculates the conditions for the solutions of the wave functions, which I can follow perfectly.
At that point...
Hey,
We are given the 1s spatial wave function for the hydrogen atom:
\psi(\vec{r}) = \frac{1}{\sqrt{a_{0}^3r}}e^{-r/a_{0}
We are asked to find the momentum space wave function \phi(\vec{p}). Obviously this is just the Fourier transform of the spatial wave function. In calculating...
I have to show that if a wave function (Schrodinger) has a potential V(x) and the wave function's complex conjugate has a potential V'(x) and V(x) does not equal V'(x),
this contradicts the continuity equation dp/dt + div J =0
where p=charge density, and J=current density.
Can someone...
As is always my problem with physics homework, I am probably thinking to hard about this... however, I am not sure how to express this wave function!
This is the question:
24) The time independent wave function of a particle is given in the graph below. The function rises linearly from the...
Hate to ask another one of these questions, but I've just read something about the collapse of the wave function that does not seem consistent with other accounts I've read about it. From what I understand, the wave function of a system is collapsed automatically by interaction with another...
You're not understading:
Let me give you all my work to alleaviate any confusion.
Show that A = (2/L)1/2
&psi(x) = A Sin(&pi x/L)
&psi2(x) = A2 Sin2(&pi x/L)
[inte]0L &psi2dx = 1
A2[inte]0L Sin2(&pi x/L) dx = 1
Actually...
I forgot to resubsitute...
BTW: I only use a...
I assume that some speed limit must exist that limits how often we can measure something - if is exists, perhaps the Plank time unit governs this? Do we know this answer? Does this relate to the speed of quantum computers?
I was once taught that we can calculate a small but non-zero probability for "quantum leaps" for things like atoms. I have tried to review this question within the context of gas molecules and for solids, but alas, I suspect my proficiency ends with very simple models.
So first is this...