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Results tagged with quantum-mechanics
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user 232440
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.
2
votes
Accepted
Time-ordering and Minkowski metric's negative sign
There are lots of answers already on the site for example here that discuss in varying levels of detail/nuance the ways in which the metric signature changes things. The common theme among all of them …
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2
votes
Interpretation of Sakurai's ambiguous statement about wavefunctions and overlap
Consider two functions, $$f(x):\Bbb R^3\rightarrow \Bbb R$$ $$g(x):\Bbb R^3\rightarrow \Bbb R$$ They may separately have non-zero integral on all of $\Bbb R^3$, i.e.: $$\int_{\Bbb R^3}f(x)\mathrm dx\n …
- 7,008
4
votes
Hamiltonian Operator
It is not correct that applying the Hamiltonian gives you the energy of the system if the system is in a general state, $|\psi\rangle$.
Upon measurement (in an experiment for example) the system "jump …
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0
votes
1
answer
54
views
Extra second-order term in Dyson Formula Expansion in David Tong's Notes
Right at the bottom of page 52 of David Tong's QFT notes we have just defined the time ordered Dyson formula, David Tong then shows the expansion of $(3.20)$ however an extra second-order term has app …
- 7,008
10
votes
2
answers
619
views
Time evolution in quantum mechanics of states not contained in the Hilbert space
Eigenstates of, for example, $\hat p$, are not elements of the standard quantum mechanical Hilbert space, i.e. $\psi(x)=e^{ipx}\notin\mathcal L^2(\Bbb R)$. This prompts the question of - given that af …
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2
votes
Inner product of $\langle \phi | \psi \rangle$ gives a complex value - why/meaning?
As noted in Mike Stone's answer this is explained in any quantum mechanics textbook. The inner product is a complex inner product, in general:
$$\langle \phi|\psi\rangle\in \Bbb C \tag{1}.$$
In quantu …
- 7,008
1
vote
Accepted
Biconditionality of the compatibility theorem for commuting operators
Yes it is if and only if. Given two observables $X$ and $Y$:
$[X,Y]=0$
$X$ and $Y$ have a common eigenbasis
are equivalent statements.
- 7,008
4
votes
Accepted
Are particles literally waves or just abstract probability waves?
It's tempting to try to interpret the mathematics of QM or QFT too literally, however this can be a dangerous game to play. Quite often you will cause more problems for yourself than you will solve by …
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7
votes
Accepted
Correct interpretation of $\langle x | \psi \rangle$?
The first of your two suggestions doesn't make sense, "the probability amplitude of finding the particle at position $x$ in the state $|\psi\rangle$". The particle is either in the state $|\psi\rangle …
- 7,008
3
votes
Accepted
What physical quantity does the Hamiltonian operator represent?
The Hamiltonian corresponds to the energy of the system. The equation you have written is the Schrodinger equation and it tells you that the Hamiltonian is a special observable operator that dictates …
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0
votes
Is $(L^2, L_z)$ a complete set of commuting observables?
Regarding the Hydrogen atom, being in an eigenspace of both $L^2$ and $L_z$ means knowing the type of orbital the electron is in ($s$, $p$, $d$, etc.) - this gives the $l$ label - and also which speci …
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3
votes
Accepted
How to prove that $\sum_{n=1}^{\infty} | \phi_n \rangle \langle \phi_n | = \hat {I}$?
You can see that an operator of this type behaves as the identity in $\Bbb R^3$ with a basis $\{|i\rangle\}$:
$$\left(\sum_{i=1}^3|i\rangle\langle i|\right)|v\rangle=\sum_{i=1}^3|i\rangle\left(\langle …
- 7,008
17
votes
Accepted
Quantum mechanics and rigorous math
Kets
In quantum mechanics the possible states of the system are elements of a separable, projective Hilbert space (i.e. two states differing by an overall complex constant are equivalent). The kets e. …
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1
vote
Time dependent Schrödinger equation with time independent potential and separation of variables
Yes, see for example this small set of notes that outlines how the Schrodinger equation comes apart into two separate equations: $$-\frac{\hbar^2}{2m}\frac{d^2\psi(x)}{dx^2}+V(x)\psi(x)=E\psi(x) \tag{ …
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2
votes
Is there a difference between a quantum state and a quantum wave function?
Remember that - because our observables correspond to Hermitian operators acting on the Hilbert space, $\mathcal H$, containing the state vectors, $|\psi\rangle$ - the set of all eigenvectors of each …
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