Displaying 11-20 of 53 results found.
Decimal expansion of conductance quantum in units of S.
+0
0
7, 7, 4, 8, 0, 9, 1, 7, 2, 9, 8, 6, 3, 6, 5, 0, 6, 4, 6, 6, 8, 0, 8, 2, 3, 3, 2, 3, 3, 0, 8, 7, 6, 3, 9, 4, 3, 5, 8, 7, 2, 8, 6, 0, 4, 7, 6, 7, 3, 3, 7, 0, 9, 1, 9, 5, 6, 3, 8, 3, 8, 3, 0, 3, 1, 6, 0, 7, 3, 5, 4, 7, 7, 5, 7, 4, 1, 3, 9, 4, 7, 6
COMMENTS
Since the May 20, 2019 redefinition of the SI base units, this quantity became a true constant. - Sean A. Irvine, Jun 06 2019
FORMULA
G0 = 7.748091729*10^-5 S.
EXAMPLE
7.748 091 7346(25) * 10^-5 S according to CODATA 2010.
Triangle of numbers used to compute the frequencies of the spectral lines of the hydrogen atom.
+0
49
3, 8, 5, 15, 12, 7, 24, 21, 16, 9, 35, 32, 27, 20, 11, 48, 45, 40, 33, 24, 13, 63, 60, 55, 48, 39, 28, 15, 80, 77, 72, 65, 56, 45, 32, 17, 99, 96, 91, 84, 75, 64, 51, 36, 19, 120, 117, 112, 105, 96, 85, 72, 57, 40, 21
COMMENTS
The rationals r(m,n):=a(m,n)/(m^2*n^2), for m-1 >= n, else 0, are used to compute the frequencies of the spectral lines of the H-atom according to quantum theory: nu(m,n) = r(m,n)*c*R' with c*R'=3.287*10^15 s^(-1) an approximation for the Rydberg frequency. R' indicates, that the correction factor 1/(1+m_e/m_p), approximately 0.9995, with the masses for the electron and proton, has been used for the Rydberg constant R_infinity. c:=299792458 m/s is, per definition, the velocity of light in vacuo (see A003678). In order to compute the wave length of the spectral lines approximately one uses the reciprocal rationals: lambda(m,n):= c/nu(m,n) = (1/r(m,n))*91.1961 nm. 1 nm = 10^{-9} m. For the corresponding energies one uses approximately E(m,n) = r(m,n)*13.599 eV (electron Volts).
The author was inspired by Dewdney's book to compile this table and related ones.
For the approximate frequencies, energies and wavelengths of the first members of the Lyman (n=1, m>=2), Balmer (n=2, m>=3), Paschen (n=3, m>=4), Brackett (n=4, m>=5) and Pfund (n=5, m>=6) series see the W. Lang link under A120072. This triangle also has an interpretation related to particle spin. For proper offset such that T(0,0) = 3, then, where h-bar = h/(2*Pi) = A003676/ A019692 (= The Dirac constant, also known as Planck's reduced constant) and Spin(n/2) = h-bar/2*sqrt(n(n+2)), it follows that: h-bar/2*sqrt(T(r,k)) = h-bar/2*sqrt(T(r,0) - T(k-1,0)) = sqrt((Spin((r+1)/2))^2 - (Spin(k/2))^2). For example, for r = k = 4, then h-bar/2*sqrt(11) = h-bar/2*sqrt(T(4,4)) = h-bar/2*sqrt(T(4,0) - T(3,0)) = sqrt(h-bar^2/4*T(4,0) - h-bar^2/4*T(3,0)) = sqrt(h-bar^2/4*35 - h-bar^2/4*24) = sqrt((Spin((4+1)/2))^2 - (Spin(4/2))^2); 35 = 5*(5+2) & 24 = 4*(4+2). - Raphie Frank, Dec 30 2012
REFERENCES
A. K. Dewdney, Reise in das Innere der Mathematik, Birkhäuser, Basel, 2000, pp. 148-154; engl.: A Mathematical Mystery Tour, John Wiley & Sons, N.Y., 1999.
FORMULA
a(m,n) = m^2 - n^2 for m-1 >= n, otherwise 0.
G.f. for column n=1,2,...: x^(n+1)*((2*n+1)- (2*n-1)*x)/(1-x)^3.
G.f. for rationals r(m,n), n=1,2,...,10 see W. Lang link.
T(r,k) = T(r,0) - T(k-1,0), T(0,0) = 3. - Raphie Frank, Dec 27 2012
EXAMPLE
Triangle begins
[ 3];
[ 8, 5];
[15, 12, 7];
[24, 21, 16, 9];
[35, 32, 27, 20, 11];
...
MATHEMATICA
ColumnForm[Table[n^2 - k^2, {n, 2, 13}, {k, n - 1}], Center] (* Alonso del Arte, Oct 26 2011 *)
PROG
(PARI) nmax=400; a=vector(1+nmax*(nmax-1)\2); idx=1; for(n=2, nmax, for(k=1, n-1, a[idx]=n*n-k*k; idx++)) \\ Stanislav Sykora, Feb 17 2014 (PARI) T(n, k)=n^2-k^2;
for (n=1, 10, for(k=1, n-1, print1(T(n, k), ", ")));
CROSSREFS
Cf. A120072/ A120073 numerator and denominator tables for rationals r(m, n).
T(n,k) = numerator of 2*Pi*Sum_{j=0..n-k-1} ((-1)^j*n*(k + j + 2)*(n + k +j)!*(k + j)!^2)/((n - k - j - 1)!*(2*k + j + 1)!*j!*Gamma(k + j + 3/2)*Gamma(k + j + 5/2)), triangle read by rows (n >= 1, 0 <= k <= n - 1).
+0
1
32, 256, 128, 2144, 256, 4096, 1024, 512, 16384, 16384, 85088, 57088, 299008, 32768, 524288, 1172224, 809344, 21856256, 950272, 8388608, 2097152, 245600, 865792, 10231808, 557056, 15204352, 4194304, 134217728, 21696512, 15546368, 1305935872, 795410432, 134217728
COMMENTS
Let a_0 and h denote Bohr radius A003671 and Planck constant A003676, respectively. Then (T(n,k)/ A159983(n,k))*(n*a_0/h) is the expectation value of any inverse momentum function, where n and k are quantum numbers which are integers obeying n > k >= 0.
EXAMPLE
Triangle begins:
n\k | 0 1 2 3 4 5 6
---------------------------------------------------------------
1 | 32
2 | 256 128
3 | 2144 256 4096
4 | 1024 512 16384 16384
5 | 85088 57088 299008 32768 524288
6 | 1172224 809344 21856256 950272 8388608 2097152
7 | 245600 865792 10231808 557056 15204352 4194304 134217728
...
MATHEMATICA
T[n_, k_] := Numerator[2*Pi*Sum[((-1)^j*n*(k + j + 2)*(n + k +j)!*(k + j)!^2)/((n - k - j - 1)!*(2*k + j + 1)!*j!*Gamma[k + j + 3/2]*Gamma[k + j + 5/2]), {j, 0, n - k - 1}]]
Table[T[n, k], {n, 1, 20}, {k, 0, n - 1}] // Flatten
PROG
(Maxima) T(n, k) := num(2*%pi*sum(((-1)^j*n*(k + j + 2)*(n + k + j)!*(k + j)!^2)/((n - k - j - 1)!*(2*k + j + 1)!*j!*gamma(k + j + 3/2)*gamma(k + j + 5/2)), j, 0, n - k - 1))$
create_list(T(n, k), n, 1, 20, k, 0, n - 1);
T(n,k) = denominator of 2*Pi*Sum_{j=0..n-k-1} ((-1)^j*n*(k + j + 2)*(n + k +j)!*(k + j)!^2)/((n - k - j - 1)!*(2*k + j + 1)!*j!*Gamma(k + j + 3/2)*Gamma(k + j + 5/2)), triangle read by rows (n >= 1, 0 <= k <= n - 1).
+0
1
3, 15, 15, 105, 21, 525, 45, 35, 1575, 2205, 3465, 3465, 24255, 3465, 72765, 45045, 45045, 1576575, 85995, 945945, 297297, 9009, 45045, 675675, 45045, 1486485, 495495, 19324305, 765765, 765765, 80405325, 58963905, 11792781, 1738165, 65702637, 78217425
COMMENTS
Let a_0 and h denote Bohr radius A003671 and Planck constant A003676, respectively. Then ( A159982(n,k)/T(n,k))*(n*a_0/h) is the expectation value of any inverse momentum function, where n and k are quantum numbers which are integers obeying n > k >= 0.
EXAMPLE
Triangle begins:
n\k | 0 1 2 3 4 5 6
-------------------------------------------------------
1 | 3
2 | 15 15
3 | 105 21 525
4 | 45 35 1575 2205
5 | 3465 3465 24255 3465 72765
6 | 45045 45045 1576575 85995 945945 297297
7 | 9009 45045 675675 45045 1486485 495495 19324305
...
MATHEMATICA
T[n_, k_] := Denominator[2*Pi*Sum[((-1)^j*n*(k + j + 2)*(n + k +j)!*(k + j)!^2)/((n - k - j - 1)!*(2*k + j + 1)!*j!*Gamma[k + j + 3/2]*Gamma[k + j + 5/2]), {j, 0, n - k - 1}]]
Table[T[n, k], {n, 1, 20}, {k, 0, n - 1}] // Flatten
PROG
(Maxima) T(n, k) := denom(2*%pi*sum(((-1)^j*n*(k + j + 2)*(n + k + j)!*(k + j)!^2)/((n - k - j - 1)!*(2*k + j + 1)!*j!*gamma(k + j + 3/2)*gamma(k + j + 5/2)), j, 0, n - k - 1))$
create_list(T(n, k), n, 1, 20, k, 0, n - 1);
6, 6, 7, 3, 0, 7, 0, 5, 2, 1, 6, 5, 4, 3, 7, 1, 2, 7, 2, 3, 9, 6, 0, 1, 6, 3, 9, 1, 3, 8, 8, 4, 1, 9, 9, 2, 4, 3, 7, 1, 6, 6, 8, 3, 0, 0, 6, 9, 1, 8, 5, 7, 2, 6, 4, 5, 7, 9, 2, 5, 6, 5, 1, 6, 5, 9, 0, 5, 4, 1, 2, 9, 2, 0, 2, 2, 8, 0, 2, 1, 5
COMMENTS
This sequence is exactly equal to (Pi*h)/Spin(5/2), where h = Planck's Constant = A003676 and Spin(n/2) = h/(4*Pi) * sqrt(n(n+2)) = A003676/(10* A019694)) * sqrt( A005563(n)).
EXAMPLE
6.673070521654371272396016391388419924371668300691857264579256516590541...
MATHEMATICA
RealDigits[(4Pi^2)/Sqrt[35], 10, 120][[1]] (* Harvey P. Dale, Jan 26 2021 *)
REFERENCES
Spectrums, Our Mind-Boggling Universe from Infinitesimal to Infinity, David Blatner, Walker & Co, USA, 2012, p. 121.
FORMULA
T_P = m*c^2/k, where m ( A078301) is the Planck mass, c ( A003678) is the speed of light in a vacuum, and k ( A070063) is the Boltzmann constant.
EXAMPLE
Planck temperature = 1.416833(85) * 10^32 K, where values in parentheses indicate uncertainty in the 2 digits preceding the parentheses.
EXTENSIONS
NIST link updated, three terms removed and minor edits by Felix Fröhlich, Aug 03 2017
Decimal expansion of the characteristic impedance of vacuum in SI units as defined before 20 May 2019.
+0
9
3, 7, 6, 7, 3, 0, 3, 1, 3, 4, 6, 1, 7, 7, 0, 6, 5, 5, 4, 6, 8, 1, 9, 8, 4, 0, 0, 4, 2, 0, 3, 1, 9, 3, 0, 8, 2, 6, 8, 6, 2, 3, 5, 0, 8, 3, 5, 2, 4, 1, 8, 6, 5, 5, 2, 3, 2, 0, 7, 4, 6, 3, 8, 2, 9, 6, 7, 0, 7, 2, 6, 9, 2, 2, 1, 3, 0, 7, 6, 9, 8, 8, 8, 0, 1, 6, 6, 8, 7, 5, 1, 9, 9, 5, 6, 5, 5, 8, 6, 6, 9, 3, 6, 4, 8
COMMENTS
Denoted Z_0, this is a function of assigned metrological constants. Regarding these, see A003678 for general context notes, references and links. - Stanislav Sykora, Jun 16 2012
FORMULA
Z_0 = [magnetic_permeability_of_vacuum]*[speed_of_light_in_vacuum] = 10^-6 * A019694 * A003678 = Pi * 119.9169832 Ohms, where 119.9169832 = 4*10^-7 * 299792458 (exact value). [The 2nd equality is valid only for the SI definition that was in effect until 2019.] Equals 2*alpha*h/e^2, where alpha is the fine-structure constant, h is the Planck constant and e is the elementary charge. The decimal expansion of Z_0 in SI units after the redefinition in 2019 is 2* A003673* A003676/ A081823^2. - Jianing Song, Nov 25 2023
EXAMPLE
376.730313 ... Ohm [also kg * m^2 * s^-3 * A^-2].
Decimal expansion of m_e*c in SI units (kg*m/s), where m_e is the electron mass and c is the speed of light in vacuum.
+0
4
2, 7, 3, 0, 9, 2, 4, 2, 9
COMMENTS
The product m_e*c is also the denominator in the formula of the electron Compton wavelength: W_C = h/(m_e*c), where h is the Planck constant.
It appears that m_e*c is also the main constant in the formula of the relativistic momentum of the electron, if such formula is written as the product of a dimensionless factor and a constant with the same dimensions as the relativistic momentum. For instance, here we write p = [1/(c^2/v^2 - 1)^(1/2)]*m_e*c instead of the standard formula p = [1/(1 - v^2/c^2)^(1/2)]*m_e*v, where v is the speed of the electron. A trigonometric version of the formula is p = tan(x)*m_e*c hence tan(x) = p/(m_e*c) assuming that sin(x) = v/c and 0 < x < Pi/2. Also p = sinh(X)*m_e*c assuming that sin(x) = tanh(X) = v/c.
Also m_e*c is the main constant in the formula of the relativistic momentum of the electron, if such formula is written as p = [(E^2/E_0^2 - 1)^(1/2)]*m_e*c where E is the relativistic energy and E_0 is the energy at rest.
Also m_e*c is equivalent to the momentum of a photon whose energy is the same as the rest energy of an electron.
Also m_e*c is equivalent to the relativistic momentum of an electron whose velocity is equal to c/sqrt(2). For more information see A229962.
EXAMPLE
2.73092429... * 10^-22 [kilogram * meter / second].
Decimal expansion of 149896229*sqrt(2).
+0
6
2, 1, 1, 9, 8, 5, 2, 8, 0, 0, 0, 0, 3, 8, 3, 2, 3, 8, 8, 7, 3, 9, 4, 4, 1, 0, 8, 5, 9, 0, 8, 5, 4, 7, 4, 7, 2, 0, 6, 1, 3, 9, 5, 2, 7, 8, 8, 6, 3, 6, 2, 4, 6, 9, 6, 9, 8, 0, 0, 0, 3, 4, 3, 4, 6, 5, 5, 1, 8, 8, 3, 5, 4, 6, 9, 2, 9, 3, 5, 6, 4, 5, 1, 8, 0, 2, 9, 5, 8, 6, 5, 8, 4, 3, 2, 1, 5, 2, 2, 2, 1, 6, 6
COMMENTS
Also decimal expansion of the speed b = c/sqrt(2) in SI units (meter/second), where c = 299792458 (m/s) is the speed of light in vacuum ( A003678). A particle (or object) with speed b has the property that its relativistic momentum equals the momentum of a virtual photon whose energy equals the rest energy of the particle. Also its relativistic de Broglie wavelength equals the Compton wavelength for the particle and therefore equals the wavelength of the photon mentioned above.
More generally it appears that the speed b is a critical speed for several relativistic magnitudes of the particle. Explanation: consider a table of relativistic magnitudes in which every formula is written as the product of a dimensionless factor and a constant with the same dimensions as the relativistic magnitude. For instance, for the relativistic momentum we write the formula p = [1/(c^2/v^2 - 1)^(1/2)]*m_0*c instead of the standard formula p = [1/(1 - v^2/c^2)^(1/2)]*m_0*v. See below:
Table 1.
----------------------------------------------------
Relativistic
magnitude Formula
----------------------------------------------------
Speed.........: v = [v/c]*c
Group velocity: g = [v/c]*c
Length........: L = [1/γ]*L_0
Momentum......: p = [1/(c^2/v^2 - 1)^(1/2)]*m_0*c
Wavenumber....: k = [1/(c^2/v^2 - 1)^(1/2)]*m_0*c/h
Wavelength....: W = [(c^2/v^2 - 1)^(1/2)]*h/(m_0*c)
Time interval.: t = γ*t_0
Mass..........: m = γ*m_0
Energy........: E = γ*m_0*c^2
Frequency.....: f = γ*m_0*c^2/h
Phase velocity: w = [c/v]*c
Kinetic energy: K = [γ - 1]*m_0*c^2
----------------------------------------------------
Where:
v is the speed of the object or particle.
c is the speed of light in vacuum ( A003678). h is the Planck constant ( A003676). L_0 is the length at rest of the object or the length at rest of a virtual cube which contains the particle.
m_0 is the mass at rest (for the electron see A081801, for the proton see A070059). t_0 is the time interval at rest.
W is the relativistic de Broglie wavelength assuming that W = h/p.
γ = [1/(1 - v^2/c^2)^(1/2)] is the Lorentz factor.
Then table 1 can be unified as shown below:
Table 2. Table 3.
------------------------------------ -----------------
Relativistic
magnitude Formula Formula
------------------------------------ -----------------
Speed.........: v = sin(x) * c v = sin(x) * v’
Group velocity: g = sin(x) * c g = sin(x) * g’
Length........: L = cos(x) * L_0 L = cos(x) * L’
Momentum......: p = tan(x) * m_0*c p = tan(x) * p’
Wavenumber....: k = tan(x) * 1/W_C k = tan(x) * k’
Wavelength....: W = cot(x) * W_C W = cot(x) * W’
Time interval.: t = sec(x) * t_0 t = sec(x) * t’
Mass..........: m = sec(x) * m_0 m = sec(x) * m’
Energy........: E = sec(x) * E_0 E = sec(x) * E’
Frequency.....: f = sec(x) * E_0/h f = sec(x) * f’
Phase velocity: w = csc(x) * c w = csc(x) * w’
Kinetic energy: K = ese(x) * E_0 K = ese(x) * K’
------------------------------------ -----------------
Where:
E_0 = m_0*c^2 is the energy at rest (for the electron see A081816), for the proton see A230438). W_C = h/(m_0*c) is the Compton wavelength for the particle (for the electron see A230436, for the proton see A230845). ese(x) = sec(x) - 1.
Table 2 is simpler than table 1 because the relativistic factors are written as trigonometric functions of the angle x assuming that sin(x) = v/c and that 0 < x < Pi/2.
Table 3 lists the simplest formulas in which the values of the constants have been interpreted as the values of the magnitudes of a virtual photon whose energy E' = h*f' is equivalent to E_0 = m_0*c^2, the rest energy of the particle.
A visualization of the relationship between the relativistic magnitudes, the quantum constants and the trigonometric functions is obtained using the first quadrant of the trigonometric circle according to the simplest table, see below:
Table 4.
-----------------------------------
sin(x) = v/v' = g/g'
cos(x) = L/L'
tan(x) = p/p' = k/k'
cot(x) = W/W'
sec(x) = t/t' = m/m' = E/E' = f/f'
csc(x) = w/w'
ese(x) = K/K'
-----------------------------------
Finally we can write that b is a critical speed because:
If v = b, for instance, we have that:
1) v/v’ = L/L’ = sin(Pi/4) = cos(Pi/4) = 2^(1/2)/2.
2) p/p’ = W/W’ = tan(Pi/4) = cot(Pi/4) = 1.
3) E/E’ = w/w’ = sec(Pi/4) = csc(Pi/4) = 2^(1/2).
Otherwise if v < b we have that:
v/v’ < L/L’ and p/p’ < W/W’ and E/E’ < w/w’.
Otherwise if v > b we have that:
v/v’ > L/L’ and p/p’ > W/W’ and E/E’ > w/w’.
MATHEMATICA
RealDigits[149896229*Sqrt[2], 10, 100][[1]] (* G. C. Greubel, Jan 26 2018 *)
CROSSREFS
Cf. A002193, A003676, A003678, A010503, A070059, A081801, A081816, A182999, A229952, A230436, A230438, A230844, A230845, A231202, A231350.
Decimal expansion of Compton wavelength in meters.
+0
6
COMMENTS
Or decimal expansion of electron Compton wavelength in meters.
Denoted lambda_C.
Also decimal expansion of the relativistic de Broglie wavelength of an electron whose velocity is equal to c/sqrt(2), where c is the speed of light in vacuum in SI units. For more information see A229962. The Compton wavelength is the wavelength of a photon whose energy is the same as the rest energy of an electron. - Omar E. Pol, Dec 03 2013
EXAMPLE
2.4263102389(16) * 10^-12 m.
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