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.
In Griffith's page 7, the following is mentioned:
What confuses me here the most is the first sentence: "Particles have a wave nature, encoded in ##\psi##"
As far as I have understood, the square of the amplitude of the wave function gives us the probability of finding a function at a...
I am meeting the momentum space wave function ##\Phi\left(p,t\right)## in chapter 3 of Griffiths & Schroeter. I have an integral which I can integrate by parts:
$$\int_{-\infty}^{\infty}{\frac{\partial}{\partial p}\left(e^\frac{ipx}{\hbar}\right)\Phi d...
Hi guys i have this exercise:
A particle of mass m, confined in the segment -a/2 < x < a/2 by a one-dimensional infinite potential well, is in a state represented by the wave function:
1. Determine the constant N from the normalization condition.
To do this, I have to integral the square...
Hi,
I'm aware of the wave function ##\Psi## of a quantum system represents basically the "continuous components" of a quantum state (a point/vector in the infinite-dimension Hilbert space) in a basis. If we take the ##\delta(x - \bar x)## eigenfunctions as basis on Hilbert space then the wave...
In answering another question, I came across a nice paper by Weinberg:
https://www.arxiv-vanity.com/papers/hep-th/9702027/
One thing that struck me was the following comment:
'In its mature form, the idea of quantum field theory is that quantum fields are the basic ingredients of the...
If we take the wave function Ψ of a particle X that gets measured in a basis with a finite countable number of eigenvectors N, then according to MWI and myself, the interpretation suggests that we can write Ψ as the sum of projections of Ψ on the eigenvectors the following way:
Ψ= Σn |e(n)> ⊗...
In a thought experiment one could arrange synchronized clocks in an inertial frame of reference such that they show the same time when the collapse happens. Does that mean that according to the relativity of simultaneity from the perspective of an observer in relative motion to that frame the...
If I am correct, the wave function is presented as a vector in Hilbert Space. Alternatively this vector can be multiplied by the identity operator. Is there a preference for one notation or the other? Are they both possible representations of the same wave function?
Physicist Dr. Muthuna Yoganathan thought the wave function was just a calculation tool, which is the standard minimal interpretation of QM. But then she started doing her own versions of the double-slit experiment at home using a red laser, culminating in purchasing a smoke machine. With that...
Dear all,
I was reading through the book "QFT for the gifted amateur" because I'm currently working on a popular science book about symmetries. Chapter 9 is about transformations of the wave function. On page 80 the book says
It's the second equality that confuses me: doesn't the statement...
Sabine Hossenfelder implies spooky action at a distance is wrong. She says “They seem to think if you do something to one particle in an entangled pair, then that will immediately affect the other one. But this isn’t so. It’s only when you measure one particle, then you have to update the...
I'm given a wave function for an electron which is given as:
For an electron in this state the kinetic energy is being measured, where the kinetic energy operator is p^2/2m. How can I find the probability (density) that an electron is found to have kinetic energy in the interval [E, E+dE]? I...
TL;DR Summary: A completely non-scientific look at wave function, because ChatGPT persuaded me to include my thoughts somehow, some way.
Howdy, the post title, as im sure you've noticed, is a weird one, and out of place in such a location. Fitting, considering the topic, I think.
Backstory...
Say you have a simplified 1d Gaussian wave function describing location of a particle.
Many worlds says that every outcome is a separate branch. Copenhagen says you will get one of those branches.
So how many distinct positions, imaginary or real, can you generate from a fixed segment of a...
In the picture below we have two identical orbitals A and B and the system has left-right symmetry. I use the notation ##|n_{A \uparrow}, n_{A \downarrow},n_{B \uparrow},n_{B \downarrow}>## which for example ##n_{A \uparrow}## indicates the number of spin-up electrons in the orbital A. I would...
I've already calculated the total spin of the system in the addition basis:
##\ket{1 \frac{3}{2} \frac{3}{2}}; \ket{1 \frac{3}{2} \frac{-3}{2}}; \ket{1 \frac{3}{2} \frac{1}{2}}; \ket{1 \frac{3}{2} \frac{-3}{2}}; \ket{0 \frac{1}{2} \frac{1}{2}}; \ket{0 \frac{1}{2} \frac{-1}{2}}; \ket{1...
In this 2011 paper, Lundeen & colleagues used weak measurement to map both imaginary and real components of a wavefunction directly, without destroying the state.
It says: “with weak measurements, it’s possible to learn something about the wavefunction without completely destroying it”. And...
I see this written or talked about so often. Pop-sci for sure. But, whatever the wave function is, and whatever might collapse it, can we agree consciousness is not required to collapse it? I.E., the moon was there before "conscious" beings, on this planet or elsewhere, viewed it? Is at...
Why on Earth does anyone, let along Roger Penrose, think gravity might be what causes the wave function to collapse? The most basic experiment in quantum physics, the double slit experiment, shows that collapse is most closely analogous to whether or not the item at issue (for example, an...
Hi,
I have hard time to really understand what's a stationary state for a wave function.
I know in a stationary state all observables are independent of time, but is the energy fix?
Is the particle has some momentum?
If a wave function oscillates between multiple energies does it means that the...
Greetings,
is it possible to characterize a sinusoidal wave in the domain of time and then pass into the domain of movement along x direction?
I start with: a is the amplitude of the sine function and ω is the angular velocity. t is the time. I can express the angular velocity in funct. of the...
The known expression of the wave function is
where A is the amplitude, k the wave number and ω the angular velocity.
The mathematical definition of arc length for a generical function in an interval [a,b] is
where, in our sinusoidal case:
For our purpose (calculation of the length in one...
The textbook I am self studying says that the wave function for a free particle with a known momentum, on the x axis, can be given as Asin(kx) and that the particle has an equal probability of being at any point along the x axis. I understand the square of the wave function to be the probability...
The following is the wave equation from Electrodynamics: $$\frac{\partial^2 \Psi}{\partial t^2} = c^2\frac{\partial^2 \Psi}{\partial x^2}$$ Where ##\Psi## is the wave function. But because of Heisenberg's Uncertainty, physicists had to come up with another equation (the Schrodinger equation)...
So what am I doing wrong here? I can clearly observe it, I'm nearly sure I can tell which particles are going throw each slit if I used another laser too. My suspicion is that the electrical current of the photon detector that uses germanium or silicon to detect the particles are influencing the...
Is there a general expression for the wave function $\psi$, which describes the electronic properties of an arbitrary covalent bond? For example is it equal to some sort of trigonometric expression?
hi guys
I am recently taking a Nuclear structure course, and have a lot of questions regarding the nuclear rotor model.
in most nuclear physics books the I have, the wave function associated with the rotor model of the nucleus is written in terms of the Wigner D functions , like the expression...
Given a wavefunction ψ(x, 0) of a free particle at initial time t=0, I need to write the general expression of the function at time t. I used a Fourier transform of ψ(x, t) in terms of ψ(p, t), but, i don't understand how to use green's functions and the time dependent schrodinger equation to...
It is asking to derive the time-independent wave function and has managed to get the answer of
and i am very confused as where (ix/a) and (-x^2/2a) came from ?
Thanks.
I was wondering, if we have a measurement of, say, spin of an electron, which can yield spin-up and spin-down in the context of MWI, then the electron gets entangled with the measurement device, yielding the wave function ##|Measurement_{spin-up}, Value_{spin-up} \rangle +...
I'm trying to find the wavelength. However, I don't understand why the wavelength is different if the wave is moving in the +z direction.
I have
##\Psi(z=15cm,t) = \hat{x} 6 cos (\frac{\pi}{3}t)##
##\Psi(z=12cm,t + 2s) = \Psi(z=18cm,t)##
For a wave moving on the -z direction
I know that the...
The final wave function solutions for a particle trapped in an infinite square well is written as:
$$\Psi(x,t) = \Sigma_{n=1}^{\infty} C_n\sqrt{\frac{2}{L_x}}sin(\frac{n\pi}{L_x}x)e^{-\frac{in^2{\pi}^2\hbar t}{2m{L_x}^2}}$$
The square of the coefficient ##C_n## i.e. ##{|C_n|}^2## is...
I first Normalise the wavefunction:
$$ \Psi_N = A*\Psi, \textrm{ where } A = (\frac{1}{\sum {|a_n^{'}|^{2}}})^{1/2} $$
$$ \Psi_N = \frac{2}{7}\phi_1^Q+\frac{3}{7}\phi_2^Q+\frac{6}{7}\phi_3^Q $$
The Eigenstate Equation is:
$$\hat{Q}\phi_n=q_n\phi_n$$
The eigenvalues are the set of possible...
Summary:: We are currently studying basics of quantum mechanics. I'm getting the theory part but it's hard to visualise everything and understand. We are given this question to plot the function so if someone could help me in this.
Plot the following function and the corresponding g²(x)
g(x)...
If ##\hat{T} = -\frac{\hbar}{2m}\frac{\mathrm{d^2} }{\mathrm{d} x^2}##, then the expectation value of the kinetic energy should be given as:
$$\begin{align*}
\left \langle T \right \rangle &= \int_{0}^{L} \sqrt{\frac{2}{L}} \sin{\left(\frac{\pi x}{L}\right)}...
Hi, I am 16 year old and I am very interested in Physics.
This summer I solved Schrödinger equation using griffiths' introduction to quantum physics and other sources. I achieved to get an exact solution of the wave function but I would like to plot it in a programm in order to get the 3d...
Hello!
Let's say we have a wave function. Maybe it's in a potential well, maybe not, I think it's arbitrary here. This wave function is one-dimensional for now to keep things simple. Then, we use a device, maybe a photon emitter and detector system where the photon crosses paths with the wave...
Existence of an universal problem solver, a polynomial-time NP-complete algorithm is a $1000000 prize question.
But suppose that we were able to know something "simple", e.g. an electron state or electron wave function exactly.
Would we be able to solve complex mathematical problems (like...
Hey there!
I have two questions regarding the Double Slit Experiment and the Wave Function Collapse.
How effective does a measuring device have to be to cause a collapse? As in, say that every second the device has a 50% chance to turn off or on for one second, does the collapse still occur...
Very early in the development of thermodynamics, it was realized that the 2nd Law of Thermodynamics is not a law fundamental to the fabric of our cosmos, but only becomes true in the limit of the number of particles. It was none other than Boltzmann himself who realized and articulated this...
(a) Let the center of the concentric spheres be the origin at ##r=0##, where r is the radius defined in spherical coordinates. The potential is given by the piece-wise function
$$V(r)=\infty, r<a$$
$$V(r)=0, a<r<R$$
$$V(r)=\infty, r<a$$
(b) we solve the Schrodinger equation and obtain...
For the sake of this question, I am primarily concerned with the position wave function. So, from my understanding, the wave function seems to 'collapse' to a few states apon measurement. We know this because, if the same particle is measured again shortly after this, it will generally remain in...
The wave function ψ(x) of a particle confined to 0 ≤ x ≤ L is given by ψ(x) = Ax, ψ(x) = 0 for x < 0 and x > L. When the wave function is normalized, the probability density at coordinate x has the value?
(A) 2x/L^2. (B) 2x^2 / L^2. (C) 2x^2 /L^3. (D) 3x^2 / L^3. (E) 3x^3 / L^3
Ans : D
I am having a trouble to understand why the helium's wave function (in which we are ignoring the electric interaction between the electrons, as well the motion and problems that arise in considering the nucleus in the wave function) can be written as the product of the wave function of both...