So last time we looked at the corner stone of Quantum Mechanics - Schrodinger equation:

* Note it's a simplified version for 1 dimension.

This time we're gonna look at some simple solutions of this equation, which will show us how microscopic particles behave.

So let's start by simplifying Schrodinger equation for a case of a free moving particle. Since the zero of potential is an arbitrary choice we choose to make U0 = 0. This gives:

(1)

Also it is useful for our solution to express the energy E in terms of momentum p (where

Substituting (2) into (1) gives:

This can be simplified further using De Broglie's wavelenght (something we covered last time:

( Schrodinger equation for a free moving particle)

So how can we solve this equation? We'll it's not a trivial process, and it doesn't really matter for learning the basics, so let's skip to the answer:

(where A is some constant, e is the exponential function, i is the square root of -1 from complex numbers, k is the wave number)

So ok, we solved the SE for a free moving particle and found the wave function, but what does it tell us about the particle? Don't worry we're gonna look at it next time. Thanks for reading!

* Note it's a simplified version for 1 dimension.

This time we're gonna look at some simple solutions of this equation, which will show us how microscopic particles behave.

So let's start by simplifying Schrodinger equation for a case of a free moving particle. Since the zero of potential is an arbitrary choice we choose to make U0 = 0. This gives:

(1)

Also it is useful for our solution to express the energy E in terms of momentum p (where

**Ek= (mv^2)/2**&**p=mv**), which gives:Substituting (2) into (1) gives:

This can be simplified further using De Broglie's wavelenght (something we covered last time:

**p=h/lambda**) and defining a quantity called wave number k:**k= 2pi/lambda:**( Schrodinger equation for a free moving particle)

So how can we solve this equation? We'll it's not a trivial process, and it doesn't really matter for learning the basics, so let's skip to the answer:

(where A is some constant, e is the exponential function, i is the square root of -1 from complex numbers, k is the wave number)

Note that this is a time-independent solution. The time dependence of the wave function comes from solving the time dependent Schrodinger equation, which gives:

So ok, we solved the SE for a free moving particle and found the wave function, but what does it tell us about the particle? Don't worry we're gonna look at it next time. Thanks for reading!

**LINK to lesson 1****LINK to lesson 2**
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