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Heisenberg Uncertainty Principle
Consider a very small particle you cannot see like an electron.
How do you know it is a particle in the first place? Could it be
something else? A wave, perhaps. If an electron were a wave,
how would it look like?
Assuming an electron is moving with a momentum p along the
x-axis.
According to de Broglie, the wavelength of the electron is λ =
The wave function of the is f(x) = sin
h
.
p
2π
x .
λ
As shown below, this wave function has a well-defined wavelength and momentum but it is
impossible to tell where the electron is.
-80 -70 -60 -50 -40
-30 -20
-10
0
10
20
30
40
50 60
70 80
How should the wave function look like then so that we can tell the electron’s position
more precisely? Perhaps a possible one would be the one shown below, where the most
disturbances appear near the origin and the electron is most likely to be around there.
-80 -70 -60 -50 -40 -30 -20 -10
0
10
20
30
40 50 60
70
80
A wave function as such requires the sum of many waves of different wavelengths.
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A wave comprising 10 different wavelengths is plotted:
f(x)
= sin(x)+sin(2x)+sin(3x)+sin(4x)+sin(5x)+sin(6x)
+sin(7x)+sin(8x)+sin(9x)+sin(10x)
A wave comprising 155 different wavelengths is plotted:
f(x)=sin(x)+sin(2x)+sin(3x)+sin(4x)+sin(5x)+sin(6x)
+sin(7x)+sin(8x)+sin(9x)+sin(10x)+sin(11x +sin(12x)
+sin(13x)+sin(14x)+sin(15x)+sin(16x)+sin(17x)+sin(18x)
+sin(19x)+sin(20x)+sin(21x)+sin(22x)+sin(23x)+sin(24x)
+sin(25x)+sin(26x)+sin(27x)+sin(28x)+sin(29x)+sin(30x)
+sin(31x)+sin(32x)+sin(33x)+sin(34x)+sin(35x)+sin(36x)
+sin(37x)+sin(38)+sin(39x)+sin(40x)+sin(41x)+sin(42x)
+sin(43x)+sin(44x)+sin(45x)+sin(46x)+sin(47x)+sin(48x)
+sin(49x)+sin(50x)+sin(51x)+sin(52x)+sin(53x)+sin(54x)
+sin(55x)+sin(56x)+sin(57x)+sin(58x)+sin(59x)+sin(60x)
+sin(61x)+sin(62x)+sin(63x)+sin(64x)+sin(65x)+sin(66x)
+sin(67x)+sin(68x)+sin(69x)+sin(70x)+sin(71x)+sin(72x)
+sin(73x)+sin(74x)+sin(75x)+sin(76x)+sin(77x)+sin(78x)
+sin(79x)+sin(80x)+sin(81x)+sin(82x)+sin(83x)+sin(84x)
+sin(85x)+sin(86x)+sin(87x)+sin(88x)+sin(89x)+sin(90x)
+sin(91x)+sin(92x)+sin(93x)+sin(94x)+sin(95x)+sin(96x)
+sin(97x)+sin(98x)+sin(99x)+sin(100x)+sin(101x)
+sin(102x)+sin(103x)+sin(104x)+sin(105x)+sin(106x)
+sin(107x)+sin(108x)+sin(109x)+sin(110x)+sin(111x)+sin(112x)+sin(113x)+sin(114x)+sin(115x)+sin(116x)+sin(117x)+sin(118x)
+sin(119x)+sin(120x)+sin(121x)+sin(122x)+sin(123x)+sin(124x)+sin(125x)+sin(126x)+sin(127x)+sin(128x)+sin(129x)+sin(130x)
+sin(131x)+sin(132x)+sin(133x)+sin(134x)+sin(135x)+sin(136x)+sin(137x)+sin(138x)+sin(140x)+sin(141x)+sin(142x)+sin(143x)
+sin(144x)+sin(145x)+sin(146x)+sin(147x)+sin(148x)+sin(149x)+sin(150x)+sin(151x)+sin(152x)+sin(153x)+sin(154x)+sin(155x)
Notice that as more wavelengths are being introduced into the wave function, the
position of the electron seems clearer.
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The electron has now become more localized but at the compromise of its wavelength,
and hence momentum.
In order to know more precisely the electron’s position, you have to compromise knowing
its wavelength. To know to know more precisely the electron’s momentum, you
compromise knowing its position.
Heisenberg’s Uncertainty Principle:
ΔxΔp ≥
h
4π
Another form of the Uncertainty Principle is shown below:
Heisenberg’s Energy–time uncertainty principle:
ΔEΔt ≥
h
4π
This principle suggests that the shorter lifetime of a state, the more uncertain we are
about the energy of the state.