logic being the same now here's the thing a system's wave function generally changes over time not always it depends on the system and its surroundings but in general the wavefunction does change how it changes or not is determined by the schrodinger equation this is the big important equation in the theory of quantum mechanics i've made lots of videos about it but if you want to get a detailed overview of the equation then please check out this video up here it basically just tells us how the wave function should change over time depending on the system and its surroundings and if the wave function changes over time then the probability of finding our particle also changes over time we can almost think of the probability as flowing through space as the wave function changes so am i seriously sitting here making this video just to tell you that the probability of finding our particle changes over time and so we should think of this as probability flowing no in reality this description of probability flowing like a fluid is even more specific but before we look at this i want to take a moment to thank the sponsor of this video ren we've all heard about the climate crisis at this point and i think that there's something that we can all do to help whether that's encouraging companies and governments to take this issue seriously or reducing our own carbon footprint so i was very happy when ren got in touch with me looking to sponsor one of my videos ren is a simple but effective way to make a difference in the climate crisis their website linked down below allows you to actually calculate your own carbon footprint and then look at different ways to reduce it you can also make a monthly contribution to offset your footprint because it goes towards funding diverse carbon reduction projects like tree planting mineral weathering and rainforest protection wren will then meticulously quantify what impact each project is having and send you monthly updates so you can be sure it's making a difference they're obviously working on lots of different projects including this particular one that i found very interesting the idea is that unused and degraded land is brought up it's then restored with techniques such as reforestation and peatland restoration at which point small bits of basil rock are spread over the forest land chemical reactions between compounds found in the basalt rock and the carbonic acid held in rainwater will then result in the formation of certain solid minerals like calcium carbonate this process is effectively trapping carbon reducing the amount of carbon dioxide in the atmosphere so i find this pretty cool so if you'd like to calculate your carbon footprint and find out more about ren then please check out the first link in the description box below it helps out the channel of course and the first NUMBER_TOKEN of you to sign up using that link will get NUMBER_TOKEN extra trees planted in your name big thanks to ren once again now let's get back to our topic at hand with a bit of mathematical manipulation we can derive an equation that looks like this looks pretty intense but here's what the terms mean rho represents the square modulus of our wave function the exact same thing we calculated earlier in order to find the probability of a particle being found in a particular region of space now remember we had to find the area under our row function between two points in space in order to work out the probability that a particle would be found between these two points if we extend this logic to multiple dimensions then we can essentially calculate the probability of finding our particle in a given volume rather than between two points on a straight line and as a result of this we can think of
START_SPONSOR_TOKEN before we look at this i want to take a moment to thank the sponsor of this video ren we've all heard about the climate crisis at this point and i think that there's something that we can all do to help whether that's encouraging companies and governments to take this issue seriously or reducing our own carbon footprint so i was very happy when ren got in touch with me looking to sponsor one of my videos ren is a simple but effective way to make a difference in the climate crisis their website linked down below allows you to actually calculate your own carbon footprint and then look at different ways to reduce it you can also make a monthly contribution to offset your footprint because it goes towards funding diverse carbon reduction projects like tree planting mineral weathering and rainforest protection wren will then meticulously quantify what impact each project is having and send you monthly updates so you can be sure it's making a difference they're obviously working on lots of different projects including this particular one that i found very interesting the idea is that unused and degraded land is brought up it's then restored with techniques such as reforestation and peatland restoration at which point small bits of basil rock are spread over the forest land chemical reactions between compounds found in the basalt rock and the carbonic acid held in rainwater will then result in the formation of certain solid minerals like calcium carbonate this process is effectively trapping carbon reducing the amount of carbon dioxide in the atmosphere so i find this pretty cool so if you'd like to calculate your carbon footprint and find out more about ren then please check out the first link in the description box below it helps out the channel of course and the first NUMBER_TOKEN of you to sign up using that link will get NUMBER_TOKEN extra trees planted in your name big thanks to ren once again now let's get back to our topic at hand END_SPONSOR_TOKEN