Suppose you want to solve Schrodinger's equation for the hydrogen atom. It's the simplest atom in the universe: only one proton and one electron, so finding the solution should be as easy as blowing up a hydrogen balloon, right? Here's an operation Schrodinger's equation requires you to perform:
Three partial derivatives with respect to spherical coordinates of the function psi. That doesn't sound too bad, but take a look at the function psi:
Obviously it's going to take you more than a minute to find a solution. Looking at the function above, it is no surprise many QM books never take you past the hydrogen atom. To find the ionization energies of all the atoms in the periodic table seems like a daunting task, requiring a supercomputer. But there is a quick-and-dirty way to achieve the task with no computer help. However, it requires that we simplify the wave function.
Often, the basis chosen for the wave function involves space coordinates, but the basis we will choose is going to be time or space-time.
Equations 1 and 2 show how a single dimension (ct or x4) subsumes all three space dimensions and time. This will allow us to do one double derivative of the function instead of the more complicated Laplacian operation.
At 3 below we express the Schrodinger equation using the space-time variable (x4).
Here is the wave function:
At 5 we choose a value for the wave number (k). At 6 we solve the kinetic-energy term and convert the units to electron volts.
At 7 through 11 we solve the potential-energy term (V):
We put all the terms together and do the math. At the other end the solution comes out (see equation 22):
Finding the ground-state ionization energy for hydrogen is now just a plug-n-chug operation:
With our new Schrodinger solution, we can easily find the ionization energy for more complicated atoms such as carbon:
The main trick is finding the values for constants A, B, and C. Here they are for atoms with up to 12 electrons:
OK, so how do we find A, B, and C? Let's take a real-world example. We want to find the values of A, B, and C for atoms with at least 13 electrons. The smallest of which is Aluminum (Al). We look at some data (click here to see the data) and choose a sample size of five elements (Z >= 13). We then make a pretty little table:
Notice as the table progresses downward it represents the equivalent of taking a double derivative of the energies. Notice at the last row, the values are pretty similar, around 3.5. Let's figure out the average:
Let's take the integral of the average with respect to Z, and then find B:
Notice we didn't use the integer Z values to find B. We used 13.5, 14.5, etc. That's because we used the numbers in the second to the last table row. If you follow the broken lines leading to the top of the table, they fall half way between the Z values. We are dealing with energy differences and not the energies themselves. Each energy difference corresponds to a half-way point.
Now that we know B, let's do another integral to find A:
Let's not forget C. This time we are dealing with the energies and not energy differences, so we plug in the integer Z values:
A particle is trapped inside a black hole. What is the probability it will escape via quantum tunneling? To figure this out, we begin by defining the variables involved:
The diagram below shows two regions: I and O. Region I is inside the event horizon, within the Schwarzshild radius (rs). Region O is outside the event horizon. The question we ask is what is the probability the particle (red dot) will reach region O? And, we could also ask what is the probability the particle will fail to reach O and remain trapped in region I? These probabilities, when added together should equal 1.
Region I covers a distance (r) from zero to rs. Region O is from rs to infinity. These will be the boundaries we will be using in the integrals below.
To find a probability in quantum mechanics we are told to square the wave-function amplitude. To see how this works, let's consider finding the probability without squaring the wave function. (See equations 1 and 2 below.)
We can model the wave function using right triangles. This is appropriate, since sine and cosine represent waves.
At the second triangle above we substitute some dummy wave functions for demonstration purposes. The trigonometric proof below shows why it is important to square a wave function to get a probability:
At equations 3 and 4 we didn't square the wave functions--they rarely add up to one, so they can't be probabilities. However once they are squared, they add up to one and could be probabilities (see equations 5 through 9). You may have noticed the wave functions have negative exponents. This feature prevents an exponential blow up to infinity.
Now, to get the right values for the probabilities we also need to include a normalization factor (A). At equations 10 to 15 below, we calculate the value of A and see why we need it. When we take the sum of all probabilities, from zero to infinity, we want the grand total to be one.
Equation 16 below is our new-and-improved wave function. Equation 17 is the one we use to calculate the probability densities of regions I and O. Equations 18 through 22 yield our desired result: equation 23, the probability density of region O; i.e., the probability that the particle will successfully escape the black hole.
Equation 24 should be the probability density for the particle's failure to escape. Let's check this:
Equation 28 confirms and matches 24. Below we use Schrodinger's equation to find the value of the wave number k:
Equation 34 shows that k is a function of V--the black hole's potential. The bigger V is, the bigger k is, the smaller the wave function and the probability that the particle will escape.
For more on the topic of quantum tunneling and QM, I highly recommend Robert Eagle's (aka: DrPhysicsA) video series:
To derive G from quantum mechanics (which covers low-energy individual particles) we begin with Schrodinger's time-independent equation:
Consider two particles with the same total energy (H), but their potential (V,V') and kinetic (h-bar terms) energies differ. We set their respective Hamiltonians (H,H') equal to each other:
Below we define their respective wave functions:
At equation 6 above we calculated the double derivative of each kinetic-energy term. Below we define the complex conjugate of each wave function. We use these complex-conjugates to find the particles' expectation values (see equation 10):
At equation 11 we equate the particles' kinetic-energy difference with their potential-energy difference. Equation 12 is a quantum-mechanical version of Einstein's field equations: The wave numbers (k,k') have the same units as spacetime curvature.
Equation 15 above is the square of the gravitational velocity (v^2) between the two particles. At equation 16 below it is perfectly legal to multiply the right side by the particles' masses (2m) and the distance (r) between said particles if we also divide by those figures. From there we can derive equation 20.
At equation 20, we can see why Newton's constant (G) is constant. Any change in the mass (m) or distance (r) between the particles causes an offsetting change in the wave-number (k) ratio; i.e., spacetime curvature changes when the distance-mass ratio changes. However, you may have wondered what would happen to G if the two particles had the same wave numbers (and the same potential and kinetic energies). The wave-number difference would be zero (or at some ground state) and so would G!
Quantum field theory to the rescue. From here on, mass (m) shall be the mass (or mass equivalent) of a field within a given volume, not an individual particle. This field could be a vacuum (not necessarily a perfect vacuum) where half-spin particles are created and annihilated. We use Dirac's equation below to assist us:
Below we derive equation 26--the square of the field's energies.
Let's suppose that this field interacts with another field, but this other field contains more mass (m') and a lower average wave number (k'). At 29 below we compare the squared energy differences of these two fields. They may or may not be equal.
Using a similar procedure we once again derive G (see equation 36):
If the two sides of the equation are not equal, we throw in a factor of n to make them so. (If the sides are equal, n = 1.)
Now Newton's constant can be constant even if the two fields have identical masses and wave numbers. A zero difference in wave numbers is offset by a zero difference in mass. (A minimum non-zero ground state also helps G to be constant.) Note that a difference in mass is the mass we measure above a given field mass. For example, when calculating Earth's gravity, the Earth's mass does not include the vacuum field mass (m). However, the total mass (m') includes the Earth and the vacuum field mass (m). Thus the Earth mass is the mass difference or net mass.
Imagine you are driving down a lonesome highway. According to relativity theory, you could look at the situation in two ways: the highway is at rest and your car is moving; or, your car is at rest and the highway is moving, so are all the trees lining the highway, so are the buildings and the sky above--the whole earth is moving while you and your car sit still and watch it all go by.
But which way of looking at the situation is true? Could they both be equally valid? No. Here's why: the amount of energy needed to move the entire earth beneath your car far exceeds the amount of energy that is in your gas tank. Thus, conservation of energy is violated if we take seriously the notion that your car is at rest and it is the earth that is moving. Therefore, it is your car that is moving relative to the earth.
Now, consider Bob and Alice. Each are inside a separate spaceship. The two spaceships are at rest relatively speaking. Both drink a potion that puts them to sleep. While they sleep, one of their spaceships is accelerated to speed v; the other remains at rest relative to the other. When Alice and Bob wake up, they must figure out which of their spaceships is going speed v. According to relativity theory, they won't be able to tell. If there is no friction and if there are no windows or portholes, both will perceive they are still at rest.
Suppose Alice and Bob can see outside, and, inside each other's ship. According to relativity theory, Alice sees Bob's clock running slower and the length of Bob's ship has shrunk. Bob, however, sees Alice's clock running slower and it is her ship that has shrunk. Is Bob right? Or is Alice? Are they both right? Are they both wrong? Did one or both of their ships shrink? Is time really running slower for Bob and/or Alice?
Since Bob sees the whole universe whisking by him, and Alice only sees Bob's ship whisking by, Chances are it's Bob's ship that is experiencing time dilation and length contraction. But how can we be sure?
Let's assume Bob and Alice each have a box that tells time by firing two photons vertically, one up and one down. The photons have a velocity c and take time t to traverse the box. The vertical distance is ct:
When Bob's ship passes Alice's here's how they see each other's clock:
They each see the other's clock lagging behind their own. When Alice notices her photons have completed the distance (ct), she looks over at Bob's and notices his photons have not. (His have gone ct'.) Bob, of course, sees the opposite.
Alice and Bob rotate their clocks 90 degrees, so that the photons go horizontally.
Again, they see their clocks going a distance of ct. When they look at the other's clock, they see this:
The above diagram indicates that time appears distorted. Going from left to right, the distance is longer and the time a photon takes to cross that distance is longer, but going right to left, distance is shorter and less time is needed. Both Alice and Bob calculate the average time and distance. They both notice the photons in both clocks, on average, complete the distance simultaneously. Since Alice and Bob have concluded the other's clock is slower, there is only one way the photons could complete their respective distances simultaneously: the other's horizontal distance has shrunk to length ct'. (In the diagram below, each assumes the other is in motion while he or she is at rest.)
However, what Alice and Bob see and conclude in the above thought experiment depends on one's confirmation bias. Instead of seeing time slow down and length contract, Alice and Bob could see the other's clock (vertical space) grow taller and the other's time in sync with their own:
If the vertical space grows taller, that explains why the other's photons take longer to complete the distance. If the horizontal space does not contract for either ship, that explains why both sets of photons completed the distance simultaneously. Also the fact that both Alice and Bob see the other's ship having the relativistic effects suggests that what they see in the other's ship is some sort of optical illusion. It ain't really happening--or is it? To know the truth about relativity we must delve deeper. Quantum physics can assist us.
Let's address whether a spaceship in motion experiences horizontal contraction or vertical growth. We can think of the particles that make up a spaceship as waves. Waves have wavelength and amplitude. We can imagine each particle's wavelength contracting. This may cause the macroscopic ship length to contract. Or, we can imagine each wave's amplitude increasing--possibly causing the ship's height to grow. We begin our investigation by defining some variables:
Here's the math that can help us figure out what is going on:
From Hooke's law we derive equation 13) above. We mentioned earlier that Bob's mass increased. Equation 13) shows that an increase in mass is a function of increased amplitude and/or decreased wavelength. So Bob's spaceship may have really shrunk horizontally and/or grown vertically. By contrast, Alice measured no increase in her mass. We can assume when Bob sees her spaceship change shape, he's seeing an illusion, like a guy looking out a train window, watching the train station go by--or, maybe he sees no change in Alice's ship.
If equation 13) is valid, we should be able to derive from it the Lorentz equation. Let's give it a go:
Equation 28) above confirms equation 13's validity. However, we still can't be sure what Alice sees when Bob's ship whizzes by. Bob's ship could have shorter length and/or taller height. Perhaps combining the Lorentz factor and quantum mechanic's wave function can help:
Take a close look at equation 37) above. Notice that the right side's numerator is cosine squared plus sine squared. This value always equals one. It is constant. This constant value represents the amplitude or height, so when velocity (v) increases, the amplitudes of all the particles in Bob's ship don't change. It's possible then that the height of Bob's ship remains constant. Only the denominator (cosine squared) can change. This represents the wavelength. So when Alice sees Bob's ship whiz by, she probably sees a shorter length and a constant height. Such a scenario is consistent with the following wavelength equation:
Equation 37b) makes no mention of wave amplitude--which implies the amplitude does not change when there's a change in momentum (p). Only the wavelength changes. Below we derive equation 41). If you check it against the Lorentz (equation 42) you will find it yields the same results--the wavelength shortens when velocity (v) increases.
The above math seems to have solved the riddle of what Alice really sees when Bob's ship flies by. However, the following math suggests that all the dimensions of Bob's ship shrink--not just the length. If we make the Lorentz equation a function of mass, the y and z dimensions shrink.
Could that be how it really works? Mass increases due to high velocity, and all three dimensions shrink? Not according to the definition of a wavelength. A wavelength is the distance along the direction of propagation ... If Bob's ship is moving along the x-axis, then we can imagine all the spaceship's particle-waves doing likewise. Thus their shortening wavelengths are along x--the ship's length. So the length of Bob's ship will likely be ct'. The height, depending on which clock is used, will be nct' or ct (nct'=ct; n is a coefficient). The width will also be some factor of ct'.
The diagrams below show the relationship between particle-waves and Bob's square clock. As velocity (v) increases, Bob's ship propagates further as expected, but the wavelengths of his ship's particles shorten, so his ship's length shortens. You can think of Bob's clock and ship as macroscopic particle waves propagating from left to right. The photon (see wavy diagonal line) is moving vertically, but its shorter wavelengths have no effect on the height, since it is the only particle moving vertically.
Equation 59) below expresses the Lorentz in terms of frequency, wavelength, mass, and time:
When energy is added, frequency increases from fo to f'. The wavelength shortens. The light-speed term (second term) stays constant, since wavelength and frequency inversely offset each other. The last term is the velocity v term. Here, mass increases, frequency increases to inversely offset the mass increase, so what's left to cause higher velocity? A shorter wavelength.
To further understand what's happening at the quantum level, we rely on a Hamiltonian (H) equation expressed in terms of mass, wavelength, amplitude, and time:
Equation 65) shows what happens to a typical particle-wave. Velocity is seen by Alice, not Bob. Bob can treat his spaceship's particles as though they are at rest. This is why the velocity term or kinetic energy term has time (t), Alice's time. Amplitude stays constant and, once again, velocity and kinetic energy are determined by wavelength. The last term is rest-mass energy. Here, when Bob's time (t') reduces, so must the wavelength (lambda): otherwise, light speed (c) would not remain constant.
After making the case above, can we conclude that Bob's spaceship actually contracts when whizzing by Alice? No. We have to take into account thermal expansion (see equations below). High velocity implies high kinetic energy; high kinetic energy implies high temperatures; high temperatures cause thermal expansion which could more than offset length contraction. At very high speeds Bob might notice he's sweating profusely. If his velocity is great enough, Alice might see a fireball going by. If she sees anything, she might presume she's at rest, since anyone moving at such high speeds might be dead.
Then again, thermal transfer to the cold surrounding space might provide Bob some relief. Equation 70) below gives us the rate of heat loss. We combine equation 70) with equations 68) and 69) to get equation 71) which tells us the thermal expansion of length L, taking into account the temperature decrease due to heat loss as well as any temperature gain due to high speeds.
Keeping relativity in mind, thermal expansion occurs when the molecules that make up Bob's spaceship move at higher velocities relative to each other. If the molecules only move in the direction Bob's ship is moving, their relative velocity to each other is zero--no thermal expansion occurs. If, on the other hand, they move in many directions, their relative velocity to each other is greater than zero--and thermal expansion could offset relativity's length contraction.
What Alice sees when Bob's spaceship flies by can vary depending on the circumstances:
