Wavefunction Properties, Normalization, and Expectation Values

We are beginning to get a glimpse of quantum mechanical principles from a rigorous, mathematical perspective. Now that we know how to use operators in conjunction with wavefunctions, let's get a better sense of what wavefunctions actually are, and how their superpositions can describe the parameters of a quantum particle. Along with this, we will learn about bra-ket notation, how to normalize wavefunctions, and also calculate both inner products and expectation values, all of which will help us get a better perspective on quantum systems.

Script by Hèctor Mas

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[jacobwilliams]

Thanks for taking the time to cover these higher-level topics. I know they don't perform as well as some of your lower-level videos, but this is incredibly helpful to college-level students such as myself whose professors glossed over most of this information.

[tonystark]

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[sathirapromodya]

You deserve a Nobel Prize in Physics for simplifying it! Thanks!

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Professor Dave, I can’t thank you enough for all the effort you put into making this content. I can assure you it‘s been very helpful for students all over the world !
I‘ve recently started a quantum chemistry course at university and came to understand it better after watching your series.
If you are interested, please make a video on experimental application of quantum mechanics (for example, Hartree-Fock approximation) or elaborate on various interpretations of quantum mechanics (I find the many-world interpretation particularly mind -blowing :). Thank you in advance!

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[lingqi]

All the videos I watched from the channel, from the mundane topics, such as cheese and olive oil, to the complex topics, such as quantum mechanics, provide accurate, explicit, and most importantly, comprehensible description and explanation. Truly one of the highest quality scientific popularization channels ever made. Well done, sir, and thank you.

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[lemonreviews]

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[tokeivo]

YouTube recommended this to me.
I saw the intro and was like: "See the previous tutorial? Nonsense! It's just a bit of math, how dense do you think I am?"
... I has been a long time since I last heard so many words, of which I could follow so few, due to my own lack of knowledge. I love it! (And will be back once I've seen the previous installments)

[BruinChang]

This helps me not only get more familiar with quantum stuff, but also integrate what I learned in wavelet transformation. Much thanks to Prof Dave.

[cgw]

Thanks for uploading your video. It was an extremely clear exposition. However, I did spot a few small inaccuracies in your video.
The wavefunction is related to the vectors of the Hilbert space by ψ(x)=<x|ψ> where |x> is a position eigenstate. The vector |ψ> does not vary with position (the information for all x is encoded into the vector, whereas in your explanation the expansion has components which seemed to depend on x, which takes uncountably many values). We can also do quantum mechanics in the momentum representation using ψ(p)=<p|ψ> where |ψ> is the same state.

Strictly speaking, you made a choice that the normalization constant, a, was real and positive when you normalized your wavefunction. Wavefunctions which differ by a constant phase factor represent the same physical state, so ψ(x) and -ψ(x), for instance, are equally good at representing a quantum state. It is for this reason that your explanation of the double slit experiment is problematic, it is not very natural to add together wavefunctions, since if ψ and φ are two wavefunctions then the wavefunction ψ+φ and ψ-φ do not represent the same physical state. After normalizing and taking the modulus we find that these combinations give entirely different predictions for the probability density for the particle hitting the screen (you slightly misspoke when you included some coefficients in front of the two wavefunctions: you referred to them as probabilities rather than amplitudes).
Incidentally, the quick way of working out the expectation of the kinetic energy is to write it is as <T>=|| φ ||²/2m where φ=pψ, this avoids computing a second derivative and is especially labour-saving when working in three dimensions.

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[arezaajouneghani]

By far the most clear explanation about this subject!

[mviews]

''quantum mechanics is a probabilistic way of viewing reality that is difficult for our classically trained brains to comprehend but nevertheless this is how the universe operates on this scale.''Amazing line Sir

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[jonocour]

i didnt get this at all until 23 minutes ago, you are terrific. I have been struggling with this formy quantum module, thank you sooo much