Physics and Chemistry Revision

Welcome to my physics, mathematics and chemistry study blog! Enjoy your stay!

[edwardspoonhands]

When doing chemistry- what is a mole exactly? I can't find a simple explanation and I'm utterly confused

It’s like a dozen, except instead of 12 it’s 602,214,090,000,000,000,000,000.

So when I say I have a dozen eggs, I have 12 eggs. When I say I have a mole of eggs, I have 602,214,090,000,000,000,000,000 eggs. 

[physicsandchemistryrevision]

Hamiltonian Mechanics

Hamilton's equations of motion

See more posts about Hamiltonian mechanics

For a Hamiltonian H, given by

where T and U are the total kinetic energy and total potential energy of the system, respectively; q is a generalised position (such as x, y, or r), and; p is a generalised momentum. Using this notation, Hamilton’s equations of motion are

Notice that q with a ‘dot’ represents a generalised velocity and p with a ‘dot’ represents a generalised force.

Equations of motion derivation

We know that the kinetic energy is classically expressed as

where

is the velocity. Recalling that p = mv, we find that

Let’s now consider the partial derivative of the Hamiltonian (in equation 1) with respect to this generalised momentum p:

Clearly, U doesn’t depend on p so does not factor into this equation. Now we may evaluate the partial derivative of the kinetic energy to find

recalling that

we have

Finally,

which we recognise as the Hamiltonian equation of motion for generalised velocity!

Now, we have a potential energy, given by

where W is the work done. Given that

i.e. Newton’s Second Law of Motion, we find that

Thus, our Hamiltonian is

which has an infinitesimal change

In the following steps we divide through by the dq element, keeping in mind that the two are linearly independent and therefore do not depend on one-another:

We recognise this as the Hamiltonian equation of motion for generalised force!

[physicsandchemistryrevision]

Have you graduated from Kent yet?

I’ve not yet graduated: I will be graduating from my masters’ year at the University of Kent in 2017. However, I am currently studying at the University of California, San Diego for the year!

[just--space]

Carina Nebula

js

[physicsandchemistryrevision]

hey! i'm doing electric power at school and i'm really confused about which equation to use when you're finding the induced emf? (-n∆flux) / (∆t) or nBA2pif???? thank you!

Hi! Both equations you mention are actually very valid to help you find the induced EMF! Since I am in the midst of making posts about electromagnetism, I’ll go through this is detail.

Although I will use both calculus with vectors and scalar, pre-calc representations, for your level I’d recommend the equations with non-infinitesimal changes (signified by an ‘approximately equal to’ ‘≈’ symbol and a ‘Δ’ symbol in place of ‘d’.)

The first equation

If we consider the first equation for induced EMF you've provided, which states that

(equation 1) where N is the total number of turns in a loop of wire, t is time and ΦB is the flux of an inductive magnetic field B through a surface S (which we can define for each situation,) given by

in which: “⋅” represents the scalar (‘dot’) product; ϕ is the angle between the field lines of the magnetic field and the normal to the surface. It is worth noting that the cos ϕ part of the expression comes directly from the scalar representation of the dot product.

This is the most general expression for the induced EMF – describing the effect a changing magnetic field (or a moving/changing surface, for that matter) has on the motion of charges through a circuit. It tells us that when we move a conducting rod through a loop of current-carrying wire, we expect to observe a current induced through the rod since lines of flux are being intersected by our rod (which is our surface, in this case.)

However, if we move our rod through the wire loop—which will, for now, be carrying a DC (i.e. direct current; non-changing) signal—and hold it at the centre, we notice this new current produced by the induced EMF will drop off in proportion with how fast we move it through the field. This is because less lines of flux are being intersected per unit time, signified by the ratio of flux change to time change. This is an important feature of electromagnetic induction because it means that if the current in the first wire is constantly changing, we need not vary the position or dimensions of our rod to allow an induced current.

The second equation

A changing field is produced using an AC circuit, which varies the voltage from a positive value to a negative value within a specific time frame, given by the frequency f of the signal which oscillates over 2π radians, allowing us to define the angular frequency ω = 2πf. This principle is vital to modern life and has applications in communications methods such as Wifi, radar, GPS and Blutooth. I hope to write a set of posts about radio communications in the future but we’ll have to wait and see whether I end up having time!

We can visualise the inductive effects of an alternating current by considering how it is principally generated – using a rotating magnet in a solenoid of N turns (a solenoid is long of wire coiled multiple times to form a sort of wire tube.)

Let’s define our flux surface as the area of a wire loop so we can see the how the flux of the magnetic field through the solenoid. To simplify the situation, we’ll ignore the integration and say that S := A, where A is the cross-sectional area of the solenoid, which means that

It is worth noting the time-dependency of the angle here: our magnetic field source is now rotating, so the angle between the field lines and the surface normal will be changing with time, too! Thus, with a little bit of thinking, we can determine ϕ(t).

We know our magnet, which is producing field lines through its poles, is rotating with an angular frequency ω = 2πf and we know that f = 1∕T, where T is the time period, so

This may help us conceptually when considering the timing of the magnet’s angle, since we can express the time in terms of the total period of the motion.

Let’s say at time t = 0, the magnet has its North pole towards the surface; the angle here is zero, meaning the induced EMF is at a maximum.

One complete revolution later, where t = T, the magnet has returned to its original position and now has ϕ(T) = 2π = 0. Therefore, by substituting the current time for our period we find that

Through rearrangement (multiply both sides by t,) this becomes

and, since we know that ϕ(T) = 2π

and we know that ϕ(T) = ϕ(t), so finally:

This is our time-dependent angle, discovered through inference alone! We can check this for any stage rotation in terms of the period and will find the correct angle. This expression throughout in oscillations throughtout physics.

So,

Now we have to use a little calculus. Recall our original expression for the induced EMF, given by equation 1, and substitute our new expression for the magnetic flux density:

By applying the chain rule, which states that

we find that

and so

which, when evaluated at t = 0 (or t = T, and so on) for maximum EMF, yields the equation you provided:

From following the derivation through, it can be seen that this expression outlines the maximum EMF induced in a wire which follows a regular looping pattern around a cross sectional area A (as with a solenoid of N turns) by a magnetic field density B for some alternating current (or regular oscillation) having a frequency f. This could be used for any AC-based induction, any system involving periodically rotating magnets or something involving periodic changes in solenoid dimensions. The key point here of that this equation requires periodic oscillation oscillation and, evidently, this describes a much more specific scenario than the first equation.

The mathematical derivation itself may be a little difficult to follow as it uses mathematics that you haven’t explicitly met yet but the key concepts lie in the derivation. Just try to understand the scenario outlined and the arguments made so you can know whether a scenario is applicable.

For a little more information, which should be more aimed at the specific syllabus you are studying, see the following posts:

Faraday and Lenz’s law of electromagnetic induction

Magnet falling through copper experiment

I hope this helps! Let me know if you have any more questions!

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I'm a very religious person but I absolutely love physics and I was wondering if you think it will be hard for me to do well in the science world while still having my beliefs or will my religion get in the way of my research and the science world?

I certainly do not think religious beliefs are incompatible with studying or practicing science in any sense. I once saw a comment online that said, “Science is the study of the intricacies in God’s work” and, although I have no religion, I completely agree that one can view science as either the study of nature’s work or of God’s work and the outcome be the same.

Many of the most prolific scientists throughout history have been religious, notably including Isaac Newton who wrote his most important works with a religious subtext and even published in-depth analyses of theistic scripture. It seems to be a fairly recent phenomenon to believe that one cannot be a scientist and a theist simultaneously. Even Einstein held some spiritualist beliefs, summed up by his famous quote: “Science without religion is lame; religion without science is blind.”

You may find that particular scripture disagrees with findings and, as long as you can separate that from your scientific work, that is okay. Remember that although many religion’s scripture is viewed as the word of God, it is undeniable that they were written by humans – God may be perfect but mankind is far from it. Besides, you can never trust your editor to accurately realise your work.

I recently asked a question regarding anti-theism and science on a forum, which found that a majority (from a very small sample group of scientists) believed that an aggressive opposition to theism in the name of science was a negative thing for science and for humanity. I’d like to believe that if a person’s mind simultaneously contained both spiritualistic beliefs and the cure for cancer, there would be no one telling them that they cannot pursue the latter in consequence of the former.

The first sentence of your question should read, “I’m a very religious person and I absolutely love physics,” because, in my mind, the two are not fundamentally contradictory.

Here’s a good video from TestTube Plus discussing science and religion: Why So Many Scientists Believe In God.

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So I'm a physics major, but I'm pretty low on the spectrum. My math level is only pre cal, but I never do the homework so I end up dropping it. I haven't taken a core class yet but I'm taking physics 101 in the upcoming semester. Should I just switch majors? I feel like I'm too dumb for physics.

Honestly, do the work and you should be fine.

Physics is a lot of hard work and requires dedication. It’s rarely a case of being too dumb for physics or mathematics; if you don’t feel like you’re up to the work then you should certainly switch majors.

Perhaps you should consider engineering-based subjects. They require lower-level mathematics and a lot is pre-calc.

[physicsandchemistryrevision]

so this year at school we have to choose a subject to learn. and 2 of these subjects are physics and chemistry. idk what to choose. If I choose chemistry I won't be able to learn physics and same if I choose physics. I'm good at both... Help?

As a physics student I might be a little biased in this.

Regardless, I would certainly recommend physics. It’s a fascinating and seemingly endless subject – there’s always more to learn! If you find enjoyment in mathematics (particularly calculus) then physics is the one for you!

However, if you prefer more conceptual-based learning I’d recommend chemistry.

In terms of long-term aspirations, chemistry has a much better employment potential within the field and, as far as I can remember, jobs in chemistry are much better paid – particularly if you go into medical chemistry or pharmacology.

On the other hand, physics jobs are generally much more ‘exciting.’ Especially in academic settings, you’ll have opportunities to undertake experimental projects all over the world and physics-based industries often involve lots of similar opportunities.

Remember, however, you can always specialise into a cross-over between the two later in your academic career. For example, quantum chemistry is a super exciting field that combines elements of both subjects!

Here are some similar questions I’ve been asked in the past that may be helpful to you:

http://physicsandchemistryrevision.tumblr.com/post/108650844618/im-hoping-to-take-a-degree-in-physics-soon-may-i

http://physicsandchemistryrevision.tumblr.com/post/102821571714/hello-i-was-just-wondering-if-physics-is-hard-to

Whatever choice you make, you’ll love it! I hope this helps!

Community comments:

chemistrybangbang​ said:

I have extensive experience in both areas.

I would say that physics jobs are more ‘exciting’ and accessible ONLY in academia (i.e. research jobs that go to people with PhDs). And the academic market is VERY squeezed right now. To anonymous: Do actual research on job prospects in physics, rather than ask others who have anecdotal information only.

People with physics bachelors and even masters degrees have such a great skill-set! They go into many STEM-related jobs. For example, when I worked in the chemistry sector of industry, one of our analysts was a physicist by trade. So if you want to pursue physics, think about other career areas where a physics background, not just F = ma, but analytical skills, will be an asset.

Chemists need physics. Two semesters of college physics is not just a requirement for most chemistry undergrad programs, but so much of chemistry is just applied physics. There’s an old saying that biology is really chemistry, chemistry is really physics, physics is really math, and math is really philosophy. It’s true. As a follower of both APS and ACS (look those up for more support) publications, so many APS articles are directly chemistry-related. For example, a new element (theorized, but previously not confirmed) was confirmed recently as having been made in the lab. It was published first in APS circles.

You never said what level of school you’re in. Can you take one as a requirement and one as an elective?

Are the physics and chemistry teachers equally good? Do you like one subject over the other? If you’re going to major in one of these subjects in college, I would recommend taking whichever in high school most interests you. STEM subjects are hard. That is, regardless of natural talent, they require hard work. So it is very, very important to grow and feed your passion. And if you’re not going to major in any STEM subjects, then again, take something that will interest you.

Good luck!

bnprime​ said:

it’s high school.learn physics. it’ll make your math stronger.

[physicsandchemistryrevision]

Electromagnetism

Electric Field

Cont’d from “Coulomb’s Law”

Using the Coulomb’s force for multiple charges we can find the electric field E created by a system of charges (be it discrete or continuous).

Let’s start by re-defining the electrostatic force F for a referece charge Q by

(equation 1). Using the previously determined equations describing the Coulomb force on distributions of charge, we can hence infer that for N discrete charges, the electric field vector E is such that

(equation 2), and for a continuous charge distribution,

These are obtained simply by dividing the force expressions by the reference charge Q.

Hence, it is worth noting that the electric field is independent of any reference charge Q (as outlined by equation 1), so we can find the electric field from any charge – even from a lone point charge. As a result, this is a powerful concept in electromagnetism.

Example: Single point charge.

What is the electric field surrounding a single point charge q?

Although this question may seem simple in principle, it’s worth looking at so we can better understand how a charge produces an electric field and how that electric field behaves.

Let’s first consider equation 2 and allow N = 1, thus removing the summation. Now we have

Thus, we can infer that the electric field propagates outwards from the charge in all radial direction since E ∝ 1/|r|2.

In turn, by rearranging this expression a little we find that E ∝ 1/4π|r|2, i.e. E is inversely proportional to the surface area of a sphere. This means that at every radial distance away from the point charge, the electric field spreads out over an increasing sphere of radius r.

What would this look like diagrammatically?

We can visualise this by placing a point charge (we’ll make it a positive charge, for simplicity’s sake) at a position in space and draw the sphere mentioned above around it.

Drawing evenly spaced vector arrows starting from the central point, tangential to the “surface” of the charge, and ending at the shell of the sphere.

We find that these vector arrows represent the electric field vector E since they have a magnitude proportional to |E| and a direction the same as r.

Notice how the lines become further spaced apart as the distance increases, implying the intensity (or strength) of the field becomes weaker over greater distances. Hence, the electric field vector is often referred to as the electric field strength to imply it is a function of the area it acts over.

Gravitational fields act in a similar manner.

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Electromagnetism

Coulomb’s Law

See previous posts about Coulomb's Law

Coulomb’s law describes the force F between two stationary charges Q and q separated by a displacement r. It is given as

where r with affixed hat is the unitary vector of r (i.e. the vector providing only the direction of r) and ϵ0 is the permittivity of free space (ϵ0 ≈ 8.854 187 8F m-1).

Often, Coulomb’s law is written in terms of the Coulomb constant, kC defined such that

This definition helps simplify a lot of results found in electrostatics and simplifies Coulomb’s law to

though this is a result which we shall not use.

Multiple discrete charges

For a system of N discrete charges qi located at a displacement ri, we can apply the superposition principle, which mathematically states that

where Fi is the Coulomb force due to the ith charge, to determine the total force on a charge Q from each individual charge qi with which it interacts. Mathematically, this is

Continous charge distribution

In turn, we can express this for a continuum of charges (such as a charged plate in which the charges cannot be individually defined). Instead we consider an infinitesimally small area of charge dq and express F in terms of the displacement r of this charge distribution from a point charge Q.

If q is the total charge within the distribution then

Often, charge distributions may also be more conveniently expressed in terms of the charge density of a particular contour, surface or volume (referred to collectively as the “domain”).

So, our element of charge dq is placed within a domain Ω of total charge q, Thus, if we define our spatially-dependent (i.e. non-constant) charge density λ(r) as the charge per unit length/area/volume (depending on the type of domain), then

and so,

In many cases, we may have a constant charge density throughout our domain, which implies λ(r) = constant = λ, and so

Although the force exerted between particles tells us a lot about an electrostatic system, it also helps up define something a little more interesting − the electric field, which we’ll examine more closely in the next post!

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Quantum Physics

Finding the expectation value

Cont'd from "Expectation value general form"

The expectation value can be well understood by studying the mathematics applied to particular scenarios. In this post we will apply a quantum mechanic approach to find the expectation value of measurement.

Particle in a box example

Not to be confused with "Pressure of an ideal gas"

This system is often known as the infinite square well.

Consider a particle with wavefunction in position representation given by

where N is a normalisation factor, confined to a 1-dimensional box in 0 ≤ x ≤ L.

We measure its momentum – what is the expected value of this measurement?

Thus, we need to find the expectation value of momentum in the x-direction;

since we can assume that the particle is entirely contained within the range 0 ≤ x ≤ L the limits can be adjusted to reflect this,

First, we’ll find the probability; |⟨ x | ψ ⟩|2:

and since we know that ⟨ ψ | x ⟩ = ⟨ x | ψ ⟩* we can find the complex conjugate of the given expression. So, we find that

which can be multiplied by the original expression to find

and since e0 = 1,

Now we must find px. Let’s look at the eigen-equation of its operator:

whose calculation provides the eigenvalue px. We know that this operator is given by

which we can act onto the wavefunction to get

for which we can calculate the derivative

and we know that

so

implying that

Thus, combining these factors into the expectation value equation, we get

and so,

Hopefully, the square well scenario will be examined in much more detail in future posts − including solving the Schrödinger equation for different scenarios.

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What do you use to type the physics equation in your tumblr posts?

I’ve been asked this question in the past and wrote a really long answer to it before Tumblr inexplicably refreshed itself and I lost nearly 2 hours’ work (such is the reason one should always write any long text posts in a word processor and transfer it to Tumblr!), so I’ll try to answer as fully as possible.

There are 3 primary methods I have used on my blog.

Using LaTeX

See the LaTeX official project website

Recently I have been using an open-source (free!), WYSIWYM (“What You Say Is What You Mean”) typesetting language called LaTeX. It’s like a pretty basic coding language which helps produce minty-fresh-looking documents.

Let’s look at the Biot-Savart law as an example of the output:

Or this post on my blog.

My process is as follows:

Write the posts in a LaTeX document.

Compile them using TeX4ht, which produces .html documents and converts anything in an equation environment to an (annoying low-quality) image.

Copy and paste the html code into Tumblr’s HTML editor, which then destroys 99% of the original code in order to “get along with” the text editor (thanks, Tumblr!).

Replace the equation number by its relevant image.

‘Mathematically format’ any quantities in the main text.

What I mean by ‘mathematically format’ is:

Scalar quantities, including lowercase Greek letters representing scalars, should be italicised.

Vector quantities should be written in bold.

Operators (such as brackets, equals signs and the differential element “d” operator (recently, falling out of common practice)) and uppercase Greek letters should not be italicised.

Alternatively, you can use a LaTeX to image online converter such as this one, into which you can type an equation using LaTeX equation codeand it will convert it to an image for download, which you can hence insert onto your blog.

I have used this method in thispost on my blog (see the “Relativistic Correction” section).

This is useful if you don’t have many equations in your post but is time (and computer memory) consuming when bulk posting.

Non-essential steps

In any steps which involve the HTML editor, youcannot go back to rich text editor after making the changes. So always make these changes last in the editing process.

Serif (equation-style) font when viewing the post on your blog:

This helps when reading posts on your blog itself. Equations generally require a serif font as opposed to a sans-serif font (think Times New Roman compared to Arial, for example) because serif fonts generally use a whole separate character set for italicised letters, whereas sans-serif fonts usually just slant the characters. This makes serif fonts much easier to read in equations and also separates your area quantity “a” from your general sentence component: “a” general component.

Strike-through any equations (highlight and “ctrl + 6”). Note that this step is only for ease so you can select what you want in the text editor then bulk-replace the HTML tags created. Not recommended if you're already using struck-out text for another reason!

Change to the HTML editor, copy and paste into Notepad and make the following changes (use “ctrl + h” to bring up the “Find and replace” window):

→

→

Indented equations for viewing on the Tumblr dash:

Equations often need to stand out from the main text, whichis why they’re usually indented or centred on the page. Since Tumblr can’t handle this in its regular text editor we have to find a work-around to indent our equations. I use blockquotes, and since you can’t blockquote an image in the text-editor we have to bea bit cheeky and edit the HTML directly.

Repeat step 2 above and make the following replacements:

→

Additionally, on your blog theme you can include a little piece of CSS to centre any blockquotes and number them, too! I’ll show you how to do this in a little while.

Using Unicode

Unicode is the general kind of text we’re used to. This sentence is written in Unicode, even! I once used Unicode to write equationswhen the equations I required were basic enough for this method.

This technique is quite good because it means that people withslow internet or people on mobile don’t have to wait a long time for images to load but is limited by how complex your equations are (it also looks quite ugly, by comparison). It doesn’t, however, require knowledge of TeX and doesn’t necessarily require expensive software.

Again, let’s use the Biot-Savart Law as an output example:

B = μ0/4π ∮C Idℓ×r/|r|3

Code:

B = μ0/4π ∮C Idℓ×r/|r|3

Or look at this post on my blog.

I have a .txt file that you can download here which contains a lot of useful mathematical Unicode symbols and ones that aren’t there can be Googled or taken from a comprehensive Unicode list like this one. Word also has a handy list of symbols in Insert → Symbol. You can simply copy-and-paste these symbols into your word processor or the Tumblr text editor.

It’s surprising how many mathematical symbols are included in Unicode but, unfortunately, not all these symbols show up on all browsers (especially on Google Chrome, which is lagging behind even Internet Explorer in that respect!) so be careful when using them. You may have noticed that the bra and ket angle brackets in my recent posts appear as blank boxes on Chrome because of this.

Tricks

To get started load up a word processor (Microsoft Word or the free, open-source Apache OpenOffice) or use an online HTML editor (such as http://www.free-online-html-editor.com/). You may also use Tumblr’s text editor atyour own peril!

Fractions

Fractions can be produced either:

Using superscript and subscript text separated by a forward slash:

- If using Microsoft word: Use the shortcut ctrl + shft + “+” for superscript and ctrl + “+” for subscript.

- If using HTML: Use structure [num]/[den], where [num] and [den] are the numerator and denominator of the fraction, respectively. This produces:

[num]/[den]

Using multiple lines separated by em-dashes “—“, aligned using white-spaces (i.e. by hitting the space bar):

        dp   F = ——.         dt   

Powers and Indexes

Powers can be produced using the tags. For example, x squared, x2, is written in HTML as

x2

or they can be induced using the Microsoft Word shortcut ctrl + shft + "+" detailed above.

Indexes can be produced using the tags. For example, the nth iteration of x, xn, is written in HTML as

xn

or the Microsoft Word shortcut, ctrl + "+" from above.

Note that the tags represent italics.

Limits

Limits can be produced either:

Using subscript as lower limit and superscript as upper limit, following the example for fractions.

Using the limits on the lines above or below the equation, aligned using spaces.

          b        W = ∫ F dx.      a          

Using an HTML structure which aligns the text vertically.

TOPBOTTOM

This will unfortunately only work when viewed on your blog and will mess up the formatting when viewed on Tumblr’s dash and of any subsequent text-reblogs of the post.

Large operators

Large operators such as summations or integrations can be produced using HTML:

Limits-on-side:

OPERATORTOPBOTTOM

Limits-above-and-below:

OPERATORTOPBOTTOM

Again, this will only work when viewed on your blog andwill mess up the formatting on Tumblr’s dash and of any subsequent text-reblogs of the post.

This method has been used in this post (though I'm thinking about changing that because I like people to be able to reblog my posts.)

The “Blockquote method”

I use blockquotes to apply mathematical formatting to standalone Unicode equations. This method:

Indents the equations on the dashboard.

Allows me to easily apply a particular style toequations on my blog.

Point 2. may seem a little elusive. Essentially, what I havedone is provided my theme with CSS that centres any blockquotedtext and changes it to a slightly-larger, well-spaced serif font. I have also given posts equation numbering based on the number of blockquotes in a text post. The codes for this is as follows.

Blockquote CSS style:

blockquote{    padding:0.2em 0.5em 0.2em 0.5em;    border-left: 3px dotted #f3f3f3;    border-right: 3px dotted #f3f3f3;}blockquote p {    text-align: center;    font-family: Times New Roman, Times, serif, Palatino, serif;    font-size: 120%;    line-height: 125%;    padding: 1em 0em 0.25em -2em;    margin-bottom: none;    background-color: none;}

Equations numbering code:

.text-post {    position: relative; top: 0pt;    counter-reset:eq;}.text-post blockquote::after {    float:right;    margin-top:-2em;    text-align:right;    counter-increment: eq;    content: "(" counter(eq) ")";}

Note that in the main body, I use

[TEXT POST CODE]

to signify text posts in my theme.

If you make a lot of reblogs (which I generally don’t) you may not want blockquotes with a different font. If I want to use a blockquote (including on reblogs) without a serif font you will need to go into the HTML of the post and use the following code to replace all of the pre-existing

tags within any

tags:

[TEXT]

This code makes the font regular size and changes the font family to Helvetica or, failing that, a generic sans-serif font. It sort of forces the theme back to normal. Unfortunately, it will not align the text back to the left margin. You may wish to change Helvetica to whichever font your theme uses.

Using Microsoft Word

Microsoft Word also has an in-built equations editor, induced using alt + “+”, or by clicking Insert → Equation. This equations editor uses LaTeX formatting (since it basically uses a TeX engine) and can be exported into images in a similar way to TeX4ht.

Simply create your document and once finished go on File → Save As… and under “Save As Type" select “Web Page (.htm)”. This should convert equations to images.

Again you can open the web page, right click and select “View Source,” which should open up the HTML code and you can copy and paste this into the Tumblr HTML editor. You will haveto upload equation images and format the post to look how you want. See the “Using LaTeX” section formore in-Tumblr editing advice.

It’s worth noting that Microsoft Office is a costly software package, so if you’re short on cash go for the LaTeX option. I won’t suggest anything illegal here! ;)

I don’t personally use this technique because Word equationsstick out like an ugly, sore thumb but it is certainly the simplest method!

Using MathJax

See the MathJax official website and documentation

There is another way to use LaTeX on Tumblr. It's a sort of web-'app' that you can install into your blog's CSS to look for any LaTeX code and convert it to nicely formatted equations. Unfortunately it only works on your blog and viewing posts from the dashboard will show the source LaTeX code. If you were using MathJax on your blog the Biot-Savart Law would come up on the dashboard with the code

\begin{equation} \mathbf{B} = \frac{1}{4\pi\epsilon_0}\oint_C \frac{I\,\mathrm{d}{\boldsymbol\ell}\times\mathbf{\hat{r}}}{\left|\mathbf{r}\right|^3} \end{equation}

which MathJax would find and produce (view this ask on my blog to see how this looks using MathJax):

\begin{equation} \mathbf{B} = \frac{1}{4\pi\epsilon_0}\oint_C \frac{I\,\mathrm{d}{\boldsymbol\ell}\times\mathbf{\hat{r}}}{\left|\mathbf{r}\right|^3} \end{equation}

Not the most pretty thing for dashboard or mobile users!

However, installing it onto your blog means you can make quick and easy posts that are formatted beautifully. Just remember that, on Tumblr, the dashboard is where you get a vast majority of your post views and most of your social media reach is through reblogs. Reblogged posts will come through looking like the above if the person reblogging doesn't have MathJax installed on their blog.

Installing MathJax on Tumblr

Most simply, you can copy and paste this line of code between the

tags in your theme's HTML:

Here are some great guides I've found for installing MathJax on Tumblr:

angjookanazawa: How to write in LaTeX (using MathJax) in Tumblr

ernestyalumni: Testing: MathJax in latex.

math-is-beautiful: LaTeX (General guide)

I one day plan to start a sort-of 'campaign' with all the other science and maths blogs to encourage tumblr to introduce MathJax compatibility into their dashboard since it makes the lives of a whole part of the community much, much more difficult without it (hence the 2000+ word essay on ways to input equations into Tumblr!) but that's a battle for another day.

Additional Formatting Notes:

A pet peeve of mine is the use of “x” in place of themultiplication symbol ×. To keep your equations looking pretty and understandable, you can copy and paste the multiplication symbol from Google or insert it from Word. It is written as “\times” in LaTeX.

The hyphen “-“ and the subtraction symbol “−” are two different symbols. Although it’s a tiny difference, it looks way nicer to use the proper subtraction symbol.

If you can help it, never use less than “<” or greater than “>” symbols in place of left angle or right angle brackets.

I hope that helps! I have made a .pdf version which you can download here for future reference.

If anyone can think of anything I've left out or any other techniques, please let me know.

Happy equation-ing!

[physicsandchemistryrevision]

Quantum Physics

Expectation value general form

Cont'd from "Expectation value in quantum mechanics"

In turn, we can express this in a general form. If a general observable operator Ô has an eigenfunction | ψ ⟩ which produces an eigenvalue o, given the by the eigen-equation

then its projection onto the basis of | ψ ⟩, given by ⟨ ψ |, produces the expectation value for an observation of | ψ ⟩

which in turn produces the expression

often written as

in conformity with the expected value in statistical mathematics. Note that the eigenvalue o may not necessarily be independent of x, which is why we’ve left it in the integral.

This definition can be extended to consider all 3-dimensional space, where the expectation value of the measurement of an observable Ô of a free particle is found by

In the next post, we'll look at some examples of using this.

[walk0nthewildside-blog]

[physicsandchemistryrevision]

Quantum Physics

Expectation value in quantum mechanics

Cont'd from "Expectation value in statistical mathematics"

In quantum physics, the expectation value of the position—for example—of a given wavefunction ψ(x) is

where the "^" above x represents the operator of position, x is the eigenvalue of position and |ψ(x)|2 is the probability density (see posts about probability) of ψ, given by the product of ψ(x) with its complex conjugate; |ψ(x)|2 = ψ*(x)ψ(x).

This is analogous to the expectation value we see in statistical mathematics since |ψ(x)|2 = P(x) is the probability of finding ψ in state x, so

as previously determined!

We can extend this definition to incorporate Dirac notation, in that

is the expected value of finding ψ in the representation given by x, where x is an eigenvalue of the operator of x. This originates from the eigen-expressions of operators

which can be projected onto the basis state of | ψ ⟩, given by ⟨ ψ |, to get

where x is an eigenvalue and hence the right hand side (RHS) may be rearranged to get

(We'll call this equation 6). Now, through the study of orthogonality, we know that ⟨ ψ | ψ ⟩= 1, giving rise to the “expected” eigenvalue x resulting from a measurement of | ψ ⟩ in the x-representation:

Although this reveals the physical meaning behind this notation, we can learn more about the expectation value by inserting the completeness relation into equation 1.

The completeness relation states that, for completeness in orthonormal vector sets, we must have

which can be inserted just about anywhere in an equation!

Thus, if we insert it into equation 6, we find that

Since in general ⟨ ϕ | ξ ⟩* = ⟨ ξ | ϕ ⟩, we find that

We know that the properties of Dirac notation dictate that, ⟨ ξ | ϕ ⟩* ⟨ ξ | ϕ ⟩ = |⟨ ξ | ϕ ⟩|2, and so

[physicsandchemistryrevision]

Quantum Physics

Expectation value in statistical mathematics

Cont’d from “Average value“

Expectation value of a discrete, probabilistic set

In statistical mathematics, the expectation value ⟨A⟩ of a set A is given by the sum of each element an ∈ A, weighted by the individual probability P(an) of each respective element.

So, a set of N total elements, defined generally as

has an expectation value given by

Example:

Say you have a set of values: C(n) = {3, 4, 8, 9}, each with an equal probability.

By applying normalisation we find that each element of the set has a probability of P(n) = 1/4 (since there are only four elements, whose total probability should equal 1). Now, since each has an equal probability, what is the expected value that a measurement of the set would return? It would be the average of all elements in the set! The average is therefore nothing more than the expectation value for a discrete set of values with equal probability.

and since P(n) represents a constant P(n) = 1/4 for all n, it can be taken out of the expression

Here we can see that this expression is mathematically identical to that of the average (or mean) value. Let's hence evaluate this:

where C(1), C(2), ... are the respective elements of our set C. So

Expectation value of a probabilistic function

For a continuum of values—such as a function—the expectation value is defined in a similar way.

A function of x, f(x), for which each point in x has a probability P(x) and an expectation value found by the integral over all 1-dimesional space for each value of x weighted by its respective probability P(x). Mathematically, this is written

which can be extended to encapsulate 3-dimensional space

[physicsandchemistryrevision]

Quantum Physics

Average value

Cont'd from "Dirac Notation: Intrinsic properties"

Average of a function

A function f(x) has an average value ⟨f(x)⟩ found by integration over the range R occupied by the function, divided by the range it occupies:

Hence, a function (infinite or otherwise) f(x) has an average value over the range a to b of

Example:

Consider the function f(x) = cos(x) + 1. What is its average value in the range -π ≤ x ≤ +π?

First, let’s look at the function and find what we expect the average to be.

From looking at the graph we can see the average is the mid-point between the maximum and minima. Since the difference between these two points is 2, we expect the average to be ⟨f(x)⟩ = 1. Now we can carry out the calculation to see if this is true.

Recall the equation for the average of a function between limits a = −π and b = +π

Applying the sum rule for integration, we get

We can integrate each part to get

and compute the limits

and so, we find that

as expected!

Although we could have found this by inspecting the graph, this method is powerful because it allows us to find the average of any integrable function between any valid limits – including those which would be near-impossible to find by eye. 

Average of a discrete set

A discrete set S(n) of N total values sn, defined generally as

or

has an average ⟨S(n)⟩ given by

In turn, we can find the average of a range, from the ath value to the bth within this set:

where S(a ≤ n ≤ b) ⊂ S(n).

Example:

Consider the following set H of 4 discrete values

This could be the height (in feet) of 6 different people, for example. So, what is the average height ⟨H⟩ of these people?

We know that the average height of people is around 6 ft – so, if this is a wholly representative group of people we should expect the average to be around 6. However, by simply looking at the range of values within the set, we expect that the average will be slightly lower than this expectation.

Mathematically, we must hence find the average of the set H, found by

where hn is value of the nth element. So,

which is what we expected from looking at the set and also confirms our assumption that this group is not representative of the entire population due to their ’less-than-average’ mean height (5% lower than expected).

Measurement error:

It is worth noting that since we’re working with a physical quantity, ’height’, we should consider the measurement error on the value, which will be

where σ⟨H⟩ and σhn are the errors on the expectation value and measured values of the height, respectively. We can assume that the measuring device was accurate to the nearest 0.1 ft since the measurements are given to that accuracy, implying that σhn = 0.05.