Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, quantum field theory, quantum technology, and quantum information science.
Classical physics, the description of physics that existed before the theory of relativity and quantum mechanics, describes many aspects of nature at an ordinary (macroscopic) scale, while quantum mechanics explains the aspects of nature at small (atomic and subatomic) scales, for which classical mechanics is insufficient. Most theories in classical physics can be derived from quantum mechanics as an approximation valid at large (macroscopic) scale.Quantum mechanics differs from classical physics in that energy, momentum, angular momentum, and other quantities of a bound system are restricted to discrete values (quantization), objects have characteristics of both particles and waves (wave-particle duality), and there are limits to how accurately the value of a physical quantity can be predicted prior to its measurement, given a complete set of initial conditions (the uncertainty principle).
Quantum mechanics arose gradually from theories to explain observations which could not be reconciled with classical physics, such as Max Planck's solution in 1900 to the black-body radiation problem, and the correspondence between energy and frequency in Albert Einstein's 1905 paper which explained the photoelectric effect. These early attempts to understand microscopic phenomena, now known as the "old quantum theory", led to the full development of quantum mechanics in the mid-1920s by Niels Bohr, Erwin Schrödinger, Werner Heisenberg, Max Born and others. The modern theory is formulated in various specially developed mathematical formalisms. In one of them, a mathematical entity called the wave function provides information, in the form of probability amplitudes, about what measurements of a particle's energy, momentum, and other physical properties may yield.
Homework Statement
What is the best way to evaluate the rotation of a angular momentum eigenket (## | l,m \rangle ##) around an Cartesian axis?
2. The attempt at a solution
I've tried to use the method in...
Homework Statement
A cart roles down the track with an initial velocity vo. Because of falling rain, water starts filling the cart such that its mass increases linearly with time. The rain that has fallen on the track cause the wagon to experience a frictional force characterized with a...
Homework Statement
a particle of mass m moves in 1D potential V(x),which vanishes at infinity.
Ground state eigenfunction is ψ(x) = A sech(λx), A and λ are constants.
find the ground state energy eigenvalue of this system.
ans: -ħ^2*λ^2/2m
Homework Equations
<H> =E, H = Hamiltonian.
p=...
Homework Statement
Consider hermitian matrices M1, M2, M3, M4 that obey the property Mi Mj + Mj Mi = 2δij I where I is the identity matrix and i,j=1,2,3,4
a) Show that the eigenvalues of Mi=+/- 1 (Hint: Go to the eigenbasis of Mi and use the equation for i=j)
b) By considering the relation Mi...
Homework Statement
A function of a hermitian operator H can be written as f(H)=Σ (H)n with n=0 to n=∞.
When is (1-H)-1 defined?
Homework Equations
(1-x)-1 = Σ(-x)n= 1-x+x2-x3+...
The Attempt at a Solution
(1-H)-1 converges if each element of H converges in this series, that is (1-hi)-1...
I'm attempting to learn the mathematics of quantum mechanics using textbooks such as "The Principles of Quantum Mechanics" by Dirac. I'm uncertain however of how ket vectors work! Say |A> + |B> = |C>, then what does |C> please represent?
Homework Statement
Maybe I missed it, but in my notes and also in documents like (http://ocw.mit.edu/courses/physics/8-05-quantum-physics-ii-fall-2013/lecture-notes/MIT8_05F13_Chap_09.pdf) (equation 1.64), I see
$$ \vec{r}\cdot\vec{p} = -i\hbar r \frac{\partial}{\partial r} $$
Where ##r## is...
Homework Statement
At t=0 the wave function of a two-dimensional isotropic harmonic oscilator is
ψ(x,y,0)=A(4α^2 x^2+2αy+4α^2 xy-2) e^((-α^2 x^2)/2) e^((-α^2 y^2)/2)
where A its the normalization constant
In which instant. Wich values of total energy can we find and which probability...
Hello everyone,
In case of hydrogen atom, when we say spin up or spin down we refer to the z component of the spin. Why are we interested only in the z component of spin and not in the x and y components?
Thanks in advance
Did string theory come about as a possible solution to quantum mechanic's "something being in several places at the same time" paradox so to speak? I am not a math-guy or pretend to know anything about physics. I m just curious. I was thinking that what appears as something being in multiple...
I am a beginner in quantum mechanics and I am confused about the operator ΔA defined to be ΔA Ξ A - <A>. Can someone please tell me how to interpret <A>? From what I can understand, <A> is the expectation value and is defined to be <Ψ|A|Ψ>. But that is just a scalar correct? How do subtract a...
Homework Statement
Like ordinary wave, a particle’s wave function can be described as countless line of sine wave’s superposition. However it can also be clarified as vibration (the complex amplitude still follow the rule of local conservation) I think these two explanation are equal. Am I...
Hi, I am looking for textbooks in relativistic quantum mechanics and group theory.
I have just finished my undergraduate studies in Physics and am looking to specialise in theoretical high-energy physics. Therefore, textbooks in relativistic quantum mechanics and group theory suited for that...
Hello all. I had some questions on some of the specifics of matter-antimatter annihilation. I've tried looking this up but haven't had much success. If you guys know of any textbooks or journal articles that dig deep into the mechanics I'd be grateful if you'd post them.
Anyway, my basic...
Homework Statement
Linearly polarized light of wavelength 5890 A is incident normally on a birefringent crystal that has its optic axis parallel to the face of the crystal, along the x axis. If the incident light is polarized at an angle of 45° to the x and y axes, what is the probability that...
I completely have no idea what time-reversal mean.
Why does, by substituting -t into an equation and if the result is the same as the original equation, then the equation is said to be time-reversal symmetry?
Also, what does that 'symmetry' mean there? An even function?
If I have a wave function given to me in momentum space, bounded by constants, and I have to find the wave function in position space, when taking the Fourier transform, what will be my bounds in position space?
I have been recently introduced to QM and I am deeply interested in it. I have come to know that quantum cryptography, quantum computing, and quantum optics are the hot areas where research is going on. But I'm curious, is there theoretical research going on for understanding of the quantum...
Dear Physics Forum friends,
I am a college undergraduate in US with double majors in the mathematics and computer science. I have been doing research in the theoretical computer science, and I recently got interested in the quantum cryptography. Since I cannot take any physics course until...
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...
Homework Statement
My teacher made up this question, but I think there's something wrong.
Consider the wave packet in momentum representation defined by Φ(p)=N if -P/2<p<P/2 and Φ(p)=0 at any other point. Determine Ψ(x) and uncertainties Δp and Δx.
Homework Equations
Fourier trick and...
Homework Statement
Show that the maximum of the Planck energy density occurs for a wavelength of the form λmax = b/T, where T is the temperature and b is a constant that needs to be estimated. Homework Equations
Planck energy density
u (v,T) = 8πv2 / c3 * hv / ehv/kT-1
The Attempt at a...
Homework Statement
A particle is described by the state of the following wave function.
wavefunction(x,y) = 30/[(a^5)(b^5)]^1/2 * x(a-x) * b(b-y)
Homework Equations
integral from 0 to i of x^n * (1-x)^m dx = (n!m!)/(n+m+1)!
The Attempt at a Solution
I know that normalizing means taking the...
Hello everyone!
I'm trying to follow a solution to a problem from the book "Problems and Solutions on Quantum Mechanics", it's problem 1017. There's a step where they go on too fast, and I can't follow. I've posted the solution and where my problem is down below.
Homework Statement
The dynamics...
My question stems from a discussion I had with my colleague today. In Electomagnetic coupling , like in waveguide structures. We apply pertubation theory to find out the coupling between various modes that get coupled in the device.
My colleague said that the coupling interaction was...
I've just recently learned about Yves Clouder's hydrodynamics models that show quantum mechanics behaviour emerging from a vibrating fluid. As a "born-again student", this seems very exciting to me - especially in the sense that at the very least it offers a mental model that helps come to grips...
I have heard that in my next semester, our quantum mechanics teacher will be giving a great emphasis on difficult integrals with the most of them having to do with gamma functions.
Does anybody know a book(or any other source) that I can learn about and practice gamma functions integration...
Being a non- physicist, it is extremely difficult for me to conceptualize QM. I hope to get some definite help at this forum.
QM says that the sub-atomic particles exist as both waves and particles. It is not that much difficult to conceptualize.
QM also says that the sub-atomic particles can...
Derive the probability current density for a particle
in an electromagnetic field.
(I previously posted this on StackExchange. Please pardon,
but I have been spending a lot of time on this and if anyone
knows exactly what the subtle trick involved is, I
would really appreciate it.)...
Homework Statement
Two particles, their spin are 1/2.
The hamiltonian is ##H=\gamma s_1 \cdot s_2##
At t=0, the state ##|\alpha(0)>## is such as ##s_{1z}|\alpha(0)>=\hbar/2 |\alpha(0)>## and ##s_{2z}|\alpha(0)>=\hbar/2 |\alpha(0)>##. Find the state ##|\alpha(0)>##.2. The attempt at a...
Homework Statement
Suppose a linear operator L satisfies <A|L|A> = 0 for every state A. Show that then all matrix elements <B|L|A> = 0, and hence L = 0.
Homework Equations
##<A|L|A>=L_{AA} and <B|L|A>=L_{BA}##
The Attempt at a Solution
It seems very straight forward and I don't know how...
Homework Statement
The demonstration for the momentum operator in Quantum Mechanics goes something like this
<v>=\frac{d}{dt}<x>=\frac{d}{dt} \int x \Psi^* \Psi dx
and then one ends up with
<p>=m<v>=\int \Psi^* (-i \hbar \frac{d}{dx}) Psi dx
however, if you swap the congugates you get...
It's been a little bit since I have studied multi-particle quantum mechanics and I am a little rusty on the notation.
Let's say I have a wave function, that consists of the tensor product of two spaces, one for each particle moving, ##|\psi_1,\psi_2>##. Each of these particles is moving in a...
http://www.pnas.org/content/109/24/9314.full
According to the experimenters they have found out which path the photon took and still observed the interference pattern. So they know that the particle went through the left or right slit but at the same time saw the interference pattern built up...
Hi there. I had this question going around in my mind for a long time. Basically, I wanted to know if there is a need for the use of the complex field for the wave functions in quantum mechanics, or if quantum mechanics can be built with real wave functions, instead of working in the complex...
Hey!
I was working with Dirac's equation:
$$ ( i \hbar \gamma^\mu \partial_\mu - m ) \psi = 0, $$
and I found that if you work with a function that depends on the momentum, $$ \psi ( \mathbf{p} ), $$ you obtain:
$$ ( i \gamma \cdot \mathbf{p} + m ) \psi ( \mathbf{p} ) = 0. $$
The problem is...
Hi,
I am a math major, currently in my 3rd year of undergraduate studies, majoring in measure theory / probability / mathematical statistics. I am in the dubious situation that I will be taking a course on QM while having so far only studied classical mechanics (i.e. all chapters on classical...
1. Homework Statement
p: momentum
x: position
t: time
h_bar: Planck's constant
Ψ: wave function
Homework Equations
The Attempt at a Solution
I've posted a link to pictures. http://imgur.com/a/TKvUu
I'm not vera good at using LaTex yet :(
So I've shown that the wave equation satisfied the...
In 2012, experimenters showed that when two entangled photons separate and when one goes through the double slit, we can tell which slit it went through and see that they still created an interference pattern because the photon that it was entangled to tells us which slit it went through. What...
In the double slit experiment, what is the position and momentum of an electron/photon?
Is the position of the electron/photon which slit it went through? And is the momentum of the electron/photon the wave like interference pattern detected on the detector screen?
Homework Statement
I Have tried to solve a problem about infinite potential well with a delta well in the middle, but I haven't the results and so I can't check if the proceeding is wrong. I post the steps that I have followed hoping someone can help me to understand.
We have a particle in 1D...
Homework Statement
I'm trying to understand how we can find - at the first order - the energy-shift and the eigenstates in a degenerate case.
My notes aren't clear, so I have searched in the Sakurai, but the notation is different, I have read other notes but their notation is different...
Hallo everyone!
I am studying Physics at University level. This Fall I will enter the third year of my studies. I find the curriculum inadequate and thus try to learn stuff on my own.
I have already taken the basic courses in Calculus (single and multivariable), Complex Analysis (analytic...
Hi fellow physicists,
Suppose a spring with a stiffness k, is attached to wall and with the other side a block with a mass, m, a force F, then pulls the block away from the wall. How do you derive a function for acceleration of the block as a function of time, a(t)?
When trying to solve this I...