{"id":4439,"date":"2024-09-22T17:55:49","date_gmt":"2024-09-22T17:55:49","guid":{"rendered":"https:\/\/workhouse.sweetdishy.com\/?p=4439"},"modified":"2024-09-22T17:55:50","modified_gmt":"2024-09-22T17:55:50","slug":"the-uncertainty-principle","status":"publish","type":"post","link":"https:\/\/workhouse.sweetdishy.com\/index.php\/2024\/09\/22\/the-uncertainty-principle\/","title":{"rendered":"THE UNCERTAINTY PRINCIPLE"},"content":{"rendered":"\n

In 1925, German physicist Werner Heisenberg was an assistant to Niels Bohr at the Institute of Theoretical Physics at the University of Copenhagen. Heisenberg\u2019s research related to the development of mathematical operations to calculate the expected results of experiments on hydrogen atoms based only on observables\u2014that is, using only quantities that could be experimentally measured, either directly or indirectly, such as electron momentum and position and photon wavelength.<\/p>\n\n\n\n

Heisenberg noticed that the mathematical operations involving the position and momentum of the electron seemed to indicate that you could not measure both of these at the same time without a compromise in the precision of the measurements. The math required to solve Heisenberg\u2019s \u201cMatrix Mechanics\u201d baffled the physicists of the time (we certainly don\u2019t intend to put you through it), but there is an easier way of understanding this concept by considering the meaning of various wavefunctions.<\/p>\n\n\n\n

Let\u2019s take the simplest one\u2014the simple sine wave of wavelength \u03bb associated with an electron with momentum p<\/em>, for which according to de Broglie, \u03bb = h\/p<\/em>. This wave extends to infinity just oscillating as a simple sine wave.<\/p>\n\n\n\n

What does it mean if an electron has a wavefunction \u03a8 that is a simple sine that extends to infinity? Well, since |\u03a8|2<\/sup>\u00a0is the probability of finding the electron at a certain time and position, a wavefunction that is spread throughout space means that the electron could be anywhere. In other words, since the wavefunction has a very precise wavelength \u03bb, we know the electron\u2019s momentum very precisely, but as shown in\u00a0Figure 102a, we don\u2019t know where in space it is at all.<\/p>\n\n\n\n

\"\"<\/p>\n\n\n\n

The wavefunction that we need to find an electron at a specific position is one that has a high value at one specific location and is zero everywhere else. For example,\u00a0Figure 102b\u00a0shows us the probability density of an electron that can most likely be found in the vicinity of\u00a0x<\/em>. However, what would be the mathematical description of this wavefunction, and what would it mean for the momentum of the electron?<\/p>\n\n\n\n

To answer these questions, we need to know that waves can be combined to form a new wave. Just like the waves that we explored in our ripple tank (chapter 1,\u00a0Figure 3), we can take two waves and add them up to form a new wave. The addition, or\u00a0superposition<\/em>, of waves is the essence of the phenomenon of wave interference.<\/p>\n\n\n\n

Notice in\u00a0Figure 103a\u00a0what happens when we add four waves of different frequency, each of which is uniformly spread out in space. The resulting wave is no longer uniformly spread out, but rather is more concentrated in one place, making it more likely to find the electron at a specific place. As shown in\u00a0Figure 103b, adding the right component sine waves lets us produce an even more localized wavefunction that approximates the one on\u00a0Figure 102b.<\/p>\n\n\n\n

Figure 103\u00a0New wavefunctions can be created by adding various simple sine waves of different frequencies and amplitudes. (a<\/strong>) Adding a few sine waves at different frequencies can produce a wave that is more concentrated in a single place. (b<\/strong>) A much more localized wavefunction with a probability density |\u03a8|2<\/sup>\u00a0(bold dashed line) that peaks at a certain position can be produced by adding many short-wavelength sine wavefunctions together.\"\"<\/p>\n\n\n\n

The localized wavefunction includes many different wavelengths. That means that the electron has an equal number of probable momentums p<\/em>1<\/sub>, p<\/em>2<\/sub>, p<\/em>3<\/sub>, etc., since each simple sine wavefunction of wavelength \u03bbn<\/em><\/sub> that we mixed in has an associated momentum pn<\/sub><\/em> = h<\/em>\/\u03bbn<\/em><\/sub>.<\/p>\n\n\n\n

As such, a wavefunction that localizes the position of a particle very well must allow for a very large number of possible momentums for the particle. So, whenever we know the position of an electron very precisely, we lose precision in our knowledge of the electron\u2019s momentum. This trade-off between simultaneously knowing the position and momentum of a particle is known as the Uncertainty Principle<\/em>. Since quantum physics is the physics of Planck\u2019s constant, you may already have a feeling that this constant is at the very heart of the trade-off between simultaneously measuring the position and momentum of a particle, and you would be right! Heisenberg\u2019s estimate was that the product of the uncertainty<\/em> in the measurement of position \u0394x<\/em>, and the uncertainty<\/em> in the measurement of momentum \u0394p<\/em> had to be greater than Planck\u2019s constant h<\/em>. More rigorous analysis resulted in the modern mathematical statement:<\/p>\n\n\n\n

\"equation\"\/<\/figure>\n\n\n\n

Note that the uncertainties \u0394x<\/em> and \u0394p<\/em> are not the range of values that x<\/em> and p<\/em> can have, but rather the range within one standard deviation of those values.<\/p>\n\n\n\n

You will sometimes see the Uncertainty Principle stated as:<\/p>\n\n\n\n

\"equation\"\/<\/figure>\n\n\n\n

These two forms are identical, because Dirac\u2019s constant \"\" (pronounced \u201ch bar,\u201d also known as the \u201creduced Planck constant\u201d) is simply \"\" = h<\/em>\/2\u03c0. It is commonly used when frequency is expressed in terms of radians per second (\u201cangular frequency\u201d) instead of cycles per second. We don\u2019t want to confuse you with this, but rather wish to make it easy for you to understand what the funny \"\" symbol means in other books and papers on quantum physics.<\/p>\n\n\n\n

Heisenberg\u2019s Uncertainty Principle was published in a 1927 paper that argued why the position and momentum of a particle cannot both be measured exactly, at the same time, even in theory. For Heisenberg, and later for the whole physics society centered on Bohr\u2019s Copenhagen school, the very concepts of exact position and exact momentum together, in fact, had no meaning in nature.<\/p>\n\n\n\n

Please note that the uncertainty in the measurement of either position or momentum is not due to lack of accuracy of our measurement instruments. The compromise in the measurements happens in principle, as a law of nature. That is, even if your measurement instruments are infinitely precise, you cannot simultaneously know the position and momentum of a particle with absolute precision. For this reason, some people prefer to call this concept the Indeterminacy Principle<\/em>, so that the intrinsic indeterminate character of nature is implied.<\/p>\n\n\n\n

Notice, however, that the limitation is in the order of magnitude of Planck\u2019s constant h<\/em>. In fact, since we are dividing it by 4\u03c0, it is one order of magnitude lower than Planck\u2019s constant. This means that the Uncertainty Principle doesn\u2019t really affect our everyday experience, in which momentum is so much larger than Planck\u2019s constant.<\/p>\n","protected":false},"excerpt":{"rendered":"

In 1925, German physicist Werner Heisenberg was an assistant to Niels Bohr at the Institute of Theoretical Physics at the University of Copenhagen. Heisenberg\u2019s research related to the development of mathematical operations to calculate the expected results of experiments on hydrogen atoms based only on observables\u2014that is, using only quantities that could be experimentally measured, […]<\/p>\n","protected":false},"author":1,"featured_media":4193,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[516],"tags":[],"class_list":["post-4439","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-the-uncertainty-principle"],"jetpack_featured_media_url":"https:\/\/workhouse.sweetdishy.com\/wp-content\/uploads\/2024\/09\/uncertainty.png","_links":{"self":[{"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/4439","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/comments?post=4439"}],"version-history":[{"count":1,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/4439\/revisions"}],"predecessor-version":[{"id":4440,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/4439\/revisions\/4440"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/media\/4193"}],"wp:attachment":[{"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/media?parent=4439"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/categories?post=4439"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/tags?post=4439"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}