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If you do an image search for “quantum mechanics” you will find a lot of equations that contain
things which look like this or this or maybe also that. These things are what it called
the “bra-ket” notation. What does this mean? How do you calculate with it? And is
quantum mechanics really as difficult as they say? This is what we will talk about today.
I know that quantum mechanics is supposedly impossible to understand, but trust me when
I say the difficulty is not in the mathematics. The mathematics of quantum mechanics looks
more intimidating than it really is. To see how it works, let us have a look at
how physicists write wave-functions. The wave-function, to remind you, is what we use in quantum mechanics
to describe everything. There’s a wave-function for electrons, a wave-function for atoms,
a wave-function for Schrödinger’s cat, and so on.
The wave-function is a vector, just like the ones we learned about in school. In a three-dimensional
space, you can think of a vector as an arrow pointing from the origin of the coordinate
system to any point. You can choose a particularly convenient basis in that space, typically
these are three orthogonal vectors, each with a length of one. These basis vectors can be
written as columns of numbers which each have one entry that equals one and all other entries
equal to zero. You can then write an arbitrary vector as a sum of those basis vectors with
coefficients in front of them, say (2,3,0). These coefficients are just numbers and you
can collect them in one column. So far, so good.
Now, the wave-function in quantum mechanics is a vector just like that, except it’s
not a vector in the space we see around us, but a vector in an abstract mathematical thing
called a Hilbert-space. One of the most important differences between the wave-function and
vectors that describe directions in space is that the coefficients in quantum mechanics
are not real numbers but complex numbers, so they in general have a non-zero imaginary
part. These complex numbers can be “conjugated” which is usually denoted with a star superscript
and just means you change the sign of the imaginary part.
So the complex numbers make quantum mechanics different from your school math. But the biggest
difference is really just the notation. In quantum mechanics, we do not write vectors
with arrows. Instead we write them with these funny brackets.
Why? Well, for one because it’s convention. But it’s also a convenient way to keep track
of whether a vector is a row or a column vector. The ones we talked about so far are column-vectors.
If you have a row-vector instead, you draw the bracket on the other side. You have to
watch out here because in quantum mechanics, if you convert a row vector to a column vector,
you also have to take the complex conjugate of the coefficients.
This notation was the idea of Paul Dirac and is called the bra-ket notation. The left side,
the row vector, is the “bra” and the right side, the column vector, is the “ket”.
You can use this notation for example to write a scalar product conveniently as a “bra-ket”.
The scalar product between two vectors is the sum over the products of the coefficients.
Again, don’t forget that the bra-vector has complex conjugates on the coefficients.
Now, in quantum mechanics, all the vectors describe probabilities. And usually you chose
the basis in your space so that the basis vectors correspond to possible measurement
outcomes. The probability of a particular measurement outcome is then the absolute square
of the scalar product with the basis-vector that corresponds to the outcome. Since the
basis vectors are those which have only zero entries except for one entry which is equal
to one, the scalar product of a wave-function with a basis vector is just the coefficient
that corresponds to the one non-zero entry. And the probability is then the absolute square
of that coefficient. This prescription for obtaining probabilities
from the wave-function is known as “Born’s rule”, named after Max Born. And we know
that the probability to get *any measurement outcome is equal to one, which means that
that the sum over the squared scalar products with all basis vectors has to be one. But
this is just the length of the vector. So all wave-functions have length one.
The scalar product of the wave-function with a basis-vector is also sometimes called a
“projection” on that basis-vector. It is called a projection, because it’s the
length you get if you project the full wave-function on the direction that corresponds to the basis-vector.
Think of it as the vector casting a shadow. You could say in quantum mechanics we only
ever observe shadows of the wave-function.
The whole issue with the measurement in quantum mechanics is now that once you do a measurement,
and you have projected the wave-function onto one of the basis vectors, then its length
will no longer be equal to 1 because the probability of getting this particular measurement outcome
may have been smaller than 1. But! once you have measured the state, it is with probability
one in one of the basis states. So then you have to choose the measurement outcome that
you actually found and stretch the length of the vector back to 1. This is what is called
the “measurement update”.
Another thing you can do with these vectors is to multiply one with itself the other way
round, so that would be a ket-bra. What you get then is not a single number, as you would
get with the scalar product, but a matrix, each element of which is a product of coefficients
of the vectors. In quantum mechanics, this thing is called the “density matrix”,
and you need it to understand decoherence. We will talk about this some other time, so
keep the density matrix in mind.
Having said that, as much as I love doing these videos, if you really want to understand
quantum mechanics, you have to do some mathematical exercises on your own. A great place to do
this is Brilliant who have been sponsoring this video. Brilliant offers courses with
exercise sets on a large variety of topics in science and mathematics. It’s exactly
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The courses on Brilliant that will give you the required background for this video are
those on linear algebra and its applications: What is a vector, what is a matrix, what is
an eigenvalue, what is a linear transformation? That’s the key to understanding quantum
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You may think I made it look too easy, but it’s true: Quantum mechanics is pretty much
just linear algebra. What makes it difficult is not the mathematics. What makes it difficult
is how to interpret the mathematics. The trouble is, you cannot directly observe the wave-function.
But you cannot just get rid of it either; you need it to calculate probabilities. But
the measurement update has to be done instantaneously and therefore it does not seem to be a physical
process. So is the wave-function real? Or is it not? Physicists have debated this back
and forth for more than 100 years.
Thanks for watching, see you next week.