Simple question about vector spaces and bases in QM

In summary, the concept of a vector being the same in different bases is easier to understand in more typical vector spaces where it represents an arrow in Euclidean space. The vector involves a combination of coordinates and basis vectors, allowing for it to be the same in different bases. This can be shown mathematically using coordinates or graphically by imagining a vector as an arrow with magnitude and direction. However, in quantum mechanics, there is no graphical way to do dot products without choosing a basis, as vectors can be infinite-dimensional.
  • #1
Waxbear
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When reading in Griffiths and on Wikipedia about the vector space formulation of wavefunctions, i am constantly faced with the statement that a vector can be expressed in different bases, but that it's still the same vector. However, I'm having a hard time imagining what it is about a vector that makes it the same vector, independent of the base you express it in. As i see it, the base and coefficients completely describe the vector, so how can you say it's the same vector when you change the base and coefficients. In other words, what property of the vector is the same in all bases? (I guess this is more of a Linear algebra question really.)
 
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  • #2
Yes, this is something that is easier to understand in more typical vector spaces, where a vector represents an arrow in some Euclidean space. A typical way to write an arrow is in coordinate form, but this requires choosing a basis. A basis is a set of "basis vectors", call them e1 and e2. Then we can write the full vector v with coordinates x and y as v = x e1 + y e2. This expression shows that the vector v involves a combination of the coordinates and the basis vectors-- it is not either one by itself. This combination allows for a sense of v being the "same thing" in some other basis, call the other basis e'1 and e'2, if we can say x e1 + y e2 = x' e'1 + y' e'2. That's what is the "same thing" about v-- we can put an equals sign between these two coordinate expressions. (You can check that if you do a rotation to both the coordinates and the basis vectors, this equality will hold.)

So that's a mathematical way to show it using coordinates, but there's also a graphical way to see it, which is to imagine that a vector is an arrow. An arrow has magnitude and direction, so it is clear that neither of those properties care how you choose your basis vectors, and the coordinates are just the projections of the arrow onto the basis vectors. Different types of vector spaces have different ways to define what a "projection" entails, but the net result is the same-- you are projecting the "same vector" onto different basis states to get the different ways of coordinatizing that vector. The point is that the vector is not the list of numbers we use to coordinatize it, for the latter depends on the basis but the former does not. Physics usually uses vector spaces that also have a metric, so a concept of a "dot product." Then, you can say that "dot products" between two vectors always gives the same scalar, regardless of the basis used to do the dot product (indeed sometimes there is a graphical way to do the dot product without even using a basis at all). In quantum mechanics, dot products involve overlap integrals, and to my knowledge you generally need to choose a basis to calculate these. Does anyone know of a "graphical" way to do dot products in quantum mechanics without choosing a basis? (Other than the trivial case where both vectors are eigenstates of the same operator, since eigenstates are orthonormal.)
 
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  • #3
Thank your for your answer. I didn't know the equation you wrote in the first paragraph (equality between bases), very enlightening. I had the picture of the vector having a direction and magnitude regardless of base, however i was unable to imagine how you could talk about direction without referring to a certain basis. I guess it's just a shortcoming of the graphical way to imagine a vector.

I don't think there is a graphical way of doing dot products in QM, since vectors can be infinite-dimensional. Also, my book (Griffiths) uses the analogy of dot products in 2d euclidean space to introduce dot products in an arbitrary space, but the book also mentions that there is no way to graphically represent dot products in an n-dimensional Hilbert space.
 
  • #4
OK, I didn't think so.
 

1. What is a vector space in quantum mechanics?

A vector space in quantum mechanics is a mathematical concept that represents a set of vectors that can be added and multiplied by scalars. In quantum mechanics, these vectors represent the possible states of a physical system.

2. What is a basis in quantum mechanics?

A basis in quantum mechanics is a set of vectors that can be used to express any other vector in the vector space. These basis vectors are usually chosen to be orthogonal and normalized, making them mathematically convenient to work with.

3. How are vector spaces and bases used in quantum mechanics?

In quantum mechanics, vector spaces and bases are used to represent the possible states of a physical system, such as the position or momentum of a particle. These concepts are also essential in understanding the evolution of a system and calculating its properties and probabilities.

4. Can a vector space have more than one basis in quantum mechanics?

Yes, a vector space can have multiple bases in quantum mechanics. In fact, any set of linearly independent vectors in a vector space can be chosen as a basis. However, there is always a unique basis that is most commonly used for convenience and simplicity.

5. How does the concept of vector spaces and bases relate to the uncertainty principle in quantum mechanics?

The uncertainty principle in quantum mechanics states that it is impossible to know both the position and momentum of a particle with absolute certainty. This is because the position and momentum operators do not commute, meaning that their measurements are dependent on each other. The concept of vector spaces and bases helps to understand this principle by providing a mathematical framework for representing and manipulating the probabilities of these measurements.

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