Quantum Mechanics for Dummies Part 1
The Wave Function
This series is to introduce non-physicists to the intricacies of the physics world to truly understand where modern physics research is going and the impact of it to the world we know.
Basic Requisites:
- Highschool Calculus
- Differential equations and a little multivariable calculus
- Highschool Physics
Index: ☍
The Basics
Interpretation
Probability
Momentum
Uncertainty Principle
The Basics: ☍
Say we take a cart with mass m and we let it move on a horizontal track with force F(x, t).
Normally — Classical Mechanics — we would calculate F(x, t) = m d²x(t)/dt² ; momentum p = m dx(t)/dt ; velocity v = dx(t)/dt ;
potential -V = ∫F(x, t)∂x
But in quantum mechanics we describe this system very differently than classical mechanics.
Lets assume that the cart is now a particle with mass m, we use the particle wave functions Ψ(x, t) to calculate things like momentum. Ψ(x, t) is obtained from Schrödinger Equation:
ihₚᵢ ∂Ψ/∂t = −hₚᵢ²/2m ∂²Ψ/∂x² + VΨ
hₚᵢ=h/2pi where h is plank’s constant.
Ψ(x, 0) helps determine the initial condition and determines Ψ(x, t).
Interpretation: ☍
How does Ψ(x, t) describe the particle? As we know the particle moves or acts within a certain localized area while Ψ(x, t) spreads out into space since it describes a wave.
Born’s statistical interpretation gives us that |Ψ(x, t)|² is the probability of finding the particle at x at time t. This values is also known as the probability density.
This integration gives the probability of the particle being between a and b. If there is a |Ψ|² by x graph and around A there is a huge area then there is a larger possibility that the particle will be around A than B.
In QM we can never predict with certainty the outcome of an experiment, even if we know the wave function and the equation governing it. We can only predict the possibility of a result or information on the possible results of the experiment.
When we measure the particle at a certain time, for x, the act of measuring collapses the wave function forcing it to assume a measurable value. But when repeatedly measuring a particle after the initial measurement it gives us the same number since we have collapsed the wave function forcing it to take a value.
Thus the particle can be split into 2 different physical states:
- One, the particle follows Schrodinger Equation and the wave function evolves according to the equation
- Two, the particle is abruptly interrupted forcing Ψ to collapse. This represents the particle being measured while the former represents the particle before being measured.
Probability: ☍
As we know Ψ gives us the particle wave function and |Ψ|² gives the probability of Ψ at a certain position x and time t.
We know that the sum of the probabilities of a range of values like for Ψ is 1.
Thus if:
is the probability of Ψ between a and b. Which is less than 1 and is within a localized area between a and b. For Ψ, which spreads out through space from +∞ to -∞. The normalization of Ψ — meaning summation of the probabilities of each value in the set of values and equating it to 1 — can be taken as the same integral but now between +∞ and-∞ thus giving the answer as 1.
In quantum mechanics as we know we never have a certain value but usually a range of values with various probabilities. So if we take a certain localized area from a to b, we have a cart defined by Ψ(x, t). What do we do if we want to use the position to calculate say momentum or even just to find the position of the cart which keeps on changing with respect to Schrodinger’s Equation before measuring it. In quantum mechanics we use the expectation value which is also know as the average value of the quantity which is represented by ⟨x⟩.
What if we want to do it for a function of f(x)?
What about ⟨x²⟩?
Another important value in QM is the variance of distribution, this tells us how spread apart are the values from the expectation value:
Momentum: ☍
We know that from repeated measurements after the initial measurement we have the same measurement because of the collapse of Ψ after the initial measurement. Then how do we find the average of x? What we do is complete the first measurement which makes Ψ collapse, then return the particle back to its original form before the measurement and then measure again. As you keep doing this again and again you have many data points collected to have an expectation value for x.
What we do to find the momentum? We would need to calculate the velocity ⟨v⟩, which is d⟨x⟩/dt.
As we know |Ψ|² = Ψ*Ψ; where Ψ* is the conjugate of Ψ.
As we know the Schrodinger equation:
Using the equation above we can simplify to:
and then too:
Which is: ⟨p⟩/m
Thus:
So we can see that the operator x is sandwiched between the same 2 terms as in ⟨p⟩. So we can say that (hₚᵢ/i)(∂/∂x) can “represent” the momentum operator.
If we had to calculate the expectation value of a quantity Q with both momentum and position.
What about Kinetic Energy? K = 1/2 mv² = p²/2m:
What about force? ⟨F⟩ = d⟨p⟩/dt => using the equations above with also Schrödinger's equation for substitution we get =⟨∂V/∂x⟩. Where V is the potential energy.
Uncertainty Principle: ☍
If we remember from high school physics, we can define momentum as
p=h/λ.(De Broglie’s Wavelength)
Now we can apply it here. If we have a well defined wave which is spread out (σ ≫ 1), then the wavelength is large and the momentum is smaller, vice versa for an ill defined wavelength (σ ≪ 1). As we know spread is measured in x. Thus the spread in x is agnostic to the spread in momentum p.
The equation above is also known as the Heisenberg Uncertainty Principle.
This is the basics on how QM views particles and defines them with Ψ. Also calculating simple things such as force and momentum we used in classical mechanics.
