Questions tagged [quantum-mechanics]
Quantum mechanics describes the microscopic properties of nature in a regime where classical mechanics no longer applies. It explains phenomena such as the wave-particle duality, quantization of energy, and the uncertainty principle and is generally used in single-body systems. Use the quantum-field-theory tag for the theory of many-body quantum-mechanical systems.
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How can we interpret polarization and frequency when we are dealing with one single photon?
If polarization is interpreted as a pattern/direction of the electric-field in an electromagnetic wave and the frequency as the frequency of oscillation, how can we interpret polarization and ...
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Do photons truly exist in a physical sense or are they just a useful concept like $i = \sqrt{-1}$? [closed]
Reading about photons I hear different explanations like "elementary particle", "probability cloud", "energy quanta" and so forth. Since probably no one has ever seen a photon (if "seen" it supposedly ...
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Adding 3 electron spins
I've learned how to add two 1/2-spins, which you can do with C-G-coefficients. There are 4 states (one singlet, three triplet states). States are symmetric or antisymmetric and the quantum numbers ...
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Time as a Hermitian operator in quantum mechanics
In non-relativistic QM, on one hand we have the following relations:
$$\langle x | P | \psi \rangle ~=~ -i \hbar \frac{\partial}{\partial x} \psi(x),$$
$$\langle p | X | \psi \rangle ~=~ i \hbar \...
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Time-independent Schrödinger function: If the potential $V$ is even, then the wave function $\psi$ can always be taken to be either even or odd
I have done the Problem 2.1 in Griffiths' quantum mechanics,
and it seems not making sense to me.
What if the wave function isn't symmetric at all?
Then obviously the proof doesn't work. The ...
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How do we know that entanglement allows measurement to instantly change the other particle's state? [duplicate]
I have never found experimental evidence that measuring one entangled particle causes the state of the other entangled particle to change, rather than just being revealed.
Using the spin up spin down ...
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Why do we use Hermitian operators in QM?
Position, momentum, energy and other observables yield real-valued measurements. The Hilbert-space formalism accounts for this physical fact by associating observables with Hermitian ('self-adjoint') ...
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Classical limit of quantum mechanics
I have heard that one can recover classical mechanics from quantum mechanics in the limit the $\hbar$ goes to zero. How can this be done? (Ideally, I would love to see something like: as $\hbar$ ...
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Why does the classical path give the dominant contribution in the path integral?
Why is it that the classical path gives the dominant contribution in the quantum mechanical path integral? How do we understand this?
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Why can we treat quantum scattering problems as time-independent?
From what I remember in my undergraduate quantum mechanics class, we treated scattering of non-relativistic particles from a static potential like this:
Solve the time-independent Schrodinger ...
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Don't understand the integral over the square of the Dirac delta function
In Griffiths' Intro to QM [1] he gives the eigenfunctions of the Hermitian operator $\hat{x}=x$ as being
$$g_{\lambda}\left(x\right)~=~B_{\lambda}\delta\left(x-\lambda\right)$$
(cf. last formula on ...
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Do derivatives of operators act on the operator itself or are they "added to the tail" of operators?
How do derivatives of operators work? Do they act on the terms in the derivative or do they just get "added to the tail"? Is there a conceptual way to understand this?
For example: say you had the ...
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Are all scattering states un-normalizable?
I am an undergraduate studying quantum physics with the book of Griffiths. in 1-D problems, it said a free particle has un-normalizable states but normalizable states can be obtained by sum up the ...
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Why can interaction with a macroscopic apparatus, such as a Stern-Gerlach machine, sometimes not cause a measurement?
Consider a Stern-Gerlach machine that measures the $z$-component of the spin of an electron. Suppose our electron's initial state is an equal superposition of
$$|\text{spin up}, \text{going right} \...
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Why doesn't the uncertainty principle contradict the existence of definite-angular momentum states?
We know that for a position variable $x$ and momentum $p$, the uncertainties of the two quantities are bounded by
$$\Delta x \Delta p \gtrsim \hbar$$
Now, this is usually first explained with $x$ ...
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Galilean covariance of the Schrodinger equation
Is the Schrodinger equation covariant under Galilean transformations?
I am only asking this question so that I can write an answer myself with the content found here:
http://en.wikipedia.org/wiki/User:...
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Is it possible to recover Classical Mechanics from Schrödinger's equation?
Let me explain in details. Let $\Psi=\Psi(x,t)$ be the wave function of a particle moving in a unidimensional space. Is there a way of writing $\Psi(x,t)$ so that $|\Psi(x,t)|^2$ represents the ...
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Normalization of basis vectors with a continuous index?
I have an infinite basis which associates with each point, $x$, on the $x$-axis, a basis vector $|x\rangle$ such that the matrix of $|x\rangle$ is full of zeroes and a one by the $x^{\mathrm{th}}$ ...
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What's the physical meaning of the statement that "photons don't have positions"?
It's been mentioned elsewhere on this site that one cannot define a position operator for the one-photon sector of the quantized electromagnetic field, if one requires the position operator have ...
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Amplitude of an electromagnetic wave containing a single photon
Given a light pulse in vacuum containing a single photon with an energy $E=h\nu$, what is the peak value of the electric / magnetic field?
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Can a particle be physically observed inside a quantum barrier?
I understand that if a particle approaches a finite potential barrier of height $V_0$ with energy $E < V_0$, there is still a finite probability of finding the particle on the other side of the ...
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Why are eigenfunctions which correspond to discrete/continuous eigenvalue spectra guaranteed to be normalizable/non-normalizable?
These facts are taken for granted in a QM text I read. The purportedly guaranteed non-normalizability of eigenfunctions which correspond to a continuous eigenvalue spectrum is only partly justified by ...
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Heisenberg Uncertainty Principle scientific proof
Heisenberg's uncertainty principle states that:
$$\sigma(x)\sigma( p_x )\ge \frac {\hbar}{2}.$$
What is the scientific proof of this principle?
Operators Uncertainty
user8784
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Path integral formulation of quantum mechanics
I'm a mathematics student with not much background in physics. I'm interested in learning about the path integral formulation of quantum mechanics. Can anyone suggest me some books on this topic with ...
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Why are Only Real Things Measurable?
Why can't we measure imaginary numbers? I mean, we can take the projection of a complex wave to be the "viewable" part, so why are imaginary numbers given this immeasurable descriptor? Namely with ...
user24082
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What's wrong with this derivation that $i\hbar = 0$?
Let $\hat{x} = x$ and $\hat{p} = -i \hbar \frac {\partial} {\partial x}$ be the position and momentum operators, respectively, and $|\psi_p\rangle$ be the eigenfunction of $\hat{p}$ and therefore $$\...
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What causes a black-body radiation curve to be continuous?
The ideal black-body radiation curve (unlike the quantized emission seen from atomic spectra), is continuous over all frequencies. Many objects approximate ideal blackbodies and have radiation curves ...
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Is the uncertainty principle just saying something about what an observer can know or is it a fundamental property of nature?
I ask this question because I have read two different quotes on the uncertainty principle that don't seem to match very well. There are similar questions around here but I would like an explanation ...
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Is the Born rule a fundamental postulate of quantum mechanics?
Is the Born rule a fundamental postulate of quantum mechanics, or can it be inferred from unitary evolution?
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Creating a QM state of definite position in Fock space
I'm wondering if somebody could help me to finish a simple calculation. Let me first provide motivation for the question below: I would like to write a QM amplitude in the 'QFT-style', as
$$\langle \...
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Trace of a commutator is zero - but what about the commutator of $x$ and $p$?
Operators can be cyclically interchanged inside a trace:
$${\rm Tr} (AB)~=~{\rm Tr} (BA).$$
This means the trace of a commutator of any two operators is zero:
$${\rm Tr} ([A,B])~=~0.$$
But what about ...
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How does one account for the momentum of an absorbed photon?
Suppose I have an atom in its ground state $|g⟩$, and it has an excited state $|e⟩$ sitting at an energy $E_a=\hbar\omega_0$ above it. To excite the atom, one generally uses a photon of frequency $\...
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"FTL" Communication with Quantum Entanglement? [duplicate]
Can quantum entanglement make sending a message, whether audio, video, or even Morse code, instantaneous between two points (faster than it could travel normally at the speed of light)?
Let me first ...
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Relation between Wave equation of light and photon wave function?
Suppose in our double slit experimental setup with the usual notations $d,D$, we have a beam of light of known frequency $(\nu)$ and wavelength $(\lambda)$ - so we can describe it as $$ξ_0 = A\sin(kx-\...
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What is spontaneous symmetry breaking in quantum systems?
Most descriptions of spontaneous symmetry breaking, even for spontaneous symmetry breaking in quantum systems, actually only give a classical picture.
According to the classical picture, spontaneous ...
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Can we theoretically balance a perfectly symmetrical pencil on its one-atom tip?
I was asked by an undergrad student about this question. I think if we were to take away air molecules around the pencil and cool it to absolute zero, that pencil would theoretically balance.
Am I ...
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Interpretation of "transition rate" in Fermi's golden rule
This is a question I asked myself a couple of years back, and which a student recently reminded me of. My off-the-cuff answer is wrong, and whilst I can make some hand-waving responses I'd like a ...
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Why is there this relationship between quaternions and Pauli matrices?
I've just started studying quantum mechanics, and I've come across this correlation between Pauli matrices ($\sigma_i$) and quaternions which I can't grasp: namely, that $i\sigma_1$, $i\sigma_2$ and $...
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What is the math knowledge necessary for starting Quantum Mechanics?
Could someone experienced in the field tell me what the minimal math knowledge one must obtain in order to grasp the introductory Quantum Mechanics book/course?
I do have math knowledge but I must ...
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When does $\hbar \rightarrow 0$ provide a valid transition from quantum to classical mechanics? When and why does it fail?
Lets look at the transition amplitude $U(x_{b},x_{a})$ for a free particle between two points $x_{a}$ and $x_{b}$ in the Feynman path integral formulation
$U(x_{b},x_{a}) = \int_{x_{a}}^{x_{b}} \...
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When Eigenfunctions/Wavefunctions are real?
When the Hamiltonian is Hermitian(i,e. beyond the effective mass approximation), generally under which conditions the eigenfunctions/wavefunctions are real?
What happens in 1D case like the finite ...
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Why we use $L_2$ Space In QM?
I asked this question for many people/professors without getting a sufficient answer, why in QM Lebesgue spaces of second degree are assumed to be the one that corresponds to the Hilbert vector space ...
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Understanding Dirac's notation
Let's say I have eigenstates $|x\rangle$ associated with measurement of position. I know that the eigenstates corresponding to their respective eigenvalues form a basis, let's call it $A$. Now let's ...
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Why does spin have a discrete spectrum?
Why is it that unlike other quantum properties such as momentum and velocity, which usually are given through (probabilistic) continuous values, spin has a (probabilistic) discrete spectrum?
user8791
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A book on quantum mechanics supported by the high-level mathematics
I'm interested in quantum mechanics book that uses high level mathematics (not only the usual functional analysis and the theory of generalised functions but the theory of pseudodifferential operators ...
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Factor 2 in Heisenberg Uncertainty Principle: Which formula is correct?
Some websites and textbooks refer to $$\Delta x \Delta p \geq \frac{\hbar}{2}$$ as the correct formula for the uncertainty principle whereas other sources use the formula $$\Delta x \Delta p \geq \...
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Normalizing the solution to free particle Schrödinger equation
I have the one dimensional free particle Schrödinger equation
$$i\hbar \frac{\partial}{\partial t} \Psi (x,t) = -\frac{\hbar^2}{2m} \frac{\partial^2}{\partial x^2} \Psi (x,t), \tag{1}$$
with ...
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Confusion about rotations of quantum states: $SO(3)$ versus $SU(2)$
I'm trying to understand the relationship between rotations in "real space" and in quantum state space. Let me explain with this example:
Suppose I have a spin-1/2 particle, lets say an electron, ...
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How is a quantum superposition different from a mixed state?
According to Wikipedia, if a system has $50\%$ chance to be in state $\left|\psi_1\right>$ and $50\%$ to be in state $\left|\psi_2\right>$, then this is a mixed state.
Now, consider the state
$...
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Classical and quantum anomalies
I have read about anomalies in different contexts and ways. I would like to read an explanation that unified all these statements or points of view:
Anomalies are due to the fact that quantum field ...
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