Transcription
The state of any quantum system consisting of any number of particles can be represented with a vector of length 1. Though, this is not a traditional vector in three dimensional space. The vector can have any number of dimensions and the coefficient in each dimension is a complex number with both real and imaginary components.
In this first example, we have a two dimensional vector, with complex coefficients. Here, the real component of the first coefficient is shown on the red axis. The real and imaginary components of the second coefficient are shown on the two blue axes. In this first example, we only show cases where the imaginary component of the first coefficient is zero, because this animation is limited to only three spatial dimensions. The length of the blue line represents the magnitude of the second coefficient. The length of the red line represents the magnitude of the first coefficient. If we use the Pythagorean Theorem, we can see that the sum of the squares of the lengths of the red and blue lines must be equal to the square of the length of the green vector, which is 1. The green vector represents the quantum state of the system and it is represented with this symbol.
In this animation, we only show cases where the imaginary component of the first coefficient is zero, because this animation is limited to only three spatial dimensions. If we want to show cases where both coefficients have both real and imaginary components, we can do this with an animation depicting two numbers in the complex plane. Here, the red sphere represents the first complex coefficient and the blue sphere represents the second complex coefficient. The length of the red line represents the magnitude of the first coefficient. The length of the blue line represents the magnitude of the second coefficient. This animation is deceptive, because the red and blue lines are always 90 degrees to each other in a higher dimensional space. Therefore, as before, the sum of the squares of the lengths of the red and blue lines must be equal to the square of the length of the total vector, which must always be equal to 1.
Let us call the first complex coefficient C1 and call the second complex coefficient C2. The magnitude of C1 is equal to the length of the red line. The magnitude of C2 is equal to the length of the blue line. Let us signify the complex conjugate of C1 by the variable C1 with a star after it. Let us signify the complex conjugate of C2 by the variable C2 with a star after it. A complex number multiplied by its complex conjugate always results in a real number, with a magnitude equal to the square of the original magnitude. We can use this principle to calculate the length of a vector with complex components. The square of the length of the red line is equal to C1 multiplied by its complex conjugate. The square of the length of the blue line is equal to C2 multiplied by its complex conjugate. The sum of the squares of the lengths of the red and blue lines is the square of the length of the total vector. Therefore, the square of the length of the total vector can be calculated through the following equation. We can mathematically represent this equation in matrix form as follows. In quantum mechanics, we use the following notation to represent these types of relationships. This is a two dimensional vector with two complex coefficients.
Let us now consider a four dimensional vector with four complex coefficients. Each coefficient is represented by one of these four spheres. Each of these four lines need to be thought of as all being perpendicular to each other in a higher dimensional space. The length of each line represents the magnitude of each coefficient. The sums of the squares of the lengths of these four lines must be equal to the square of the length of the total vector, which is always equal to one. C1 represents the first complex coefficient. C2 represents the second complex coefficient. C3 represents the third complex coefficient. C4 represents the fourth complex coefficient. C1 with a star after it represents the complex conjugate of C1. C2 with a star after it represents the complex conjugate of C2. C3 with a star after it represents the complex conjugate of C3. C4 with a star after it represents the complex conjugate of C4. The square of the length of each line is calculated by its coefficient multiplied by its complex conjugate. The sums of the squares of the lengths of these four lines must be equal to the square of the length of the total vector. We can mathematically represent this equation in matrix form as follows. In quantum mechanics, we use the following notation to represent these types of relationships. This is a vector in four dimensions, with complex components for each of its four coefficients.
Let us now extent this concept to a vector with infinite dimensions. A continuous function of the variable X can be thought of as a vector of infinite dimensions. For each value of X, the function has a complex value, represented here by a line in the complex plane. Let us consider the squares of the lengths of each of these individual lines, which is always a real number. The squares of the lengths of each individual lines form a continuous function of X, where the value of the function is always a real number. The area under the entire curve represents the square of the length of the total vector. As always, the length of the total vector must be one. The area under this curve can be calculated through the following equation. In quantum mechanics, we use the following notation to represent this relationship.
If we have two different wavefunctions, which we think of as vectors, then this notation represents the dot product of the two vectors. This is similar to the traditional dot product of two vectors, except for the fact that when complex numbers are involved, we have to take the complex conjugate of the first number. If we are dealing with only real numbers, and both of our vectors have a length of one, then the dot product of the two vectors can be thought of as the projection of one vector on to the other. If the dot product of the two vectors is zero, then the two vectors are said to be orthogonal to each other. These principles remain true when dealing with the dot product of two quantum vectors of length 1, even if each vector represents a continuous function. Therefore, two continuous functions might or might not be thought of as orthogonal to each other.
In classical geometry, the word orthogonal means having a 90 degree angle. Though, when dealing with complex numbers, we have to adjust our intuition of this. Let us return to the two dimensional vector, where the real component of the first coefficient is shown on the red axis, and the real and imaginary components of the second coefficient are shown on the two blue axes. The red axis is orthogonal to the two blue axes. However, the two blue axes are not orthogonal to each other. Instead, the two blue axes are scalar multipliers of each other, where the scalar multiplier is the imaginary number “i”.
As we have seen, the word vector does not necessarily refer to an arrow in space. It refers to a mathematical object which obeys certain principles. A continuous quantum wave function in space, at any given moment in time, can be thought of as a vector with infinite dimensions. The advantage of thinking of wavefunctions as vectors is that it allows us to use the mathematical properties of vectors to understand quantum systems. Much more information is available in the videos "Quantum Spin", "Quantum Wavefunction Visualization", and in the other videos on this channel.