Lecture 2 | Modern Physics: Quantum Mechanics (Stanford)

Transcript

This program is brought to you by Stanford University. Please visit us at stanford.edu. The question is, can you defeat the uncertainty principle in the following way? You measure the position of an object first to a uh poor accuracy. It's somewhere's in here somewhere within some region of size. Let's call it delta. And then you wait a long long t... Read More

Summary

This video discusses linear vector spaces and their duals or conjugates, as well as the concept of inner product between vectors. It also provides examples of complex vector spaces, such as complex functions and column vectors. The properties of the inner product are described, including linearity, conjugation, and positivity. Finally, the specific inner product for functions and column vectors is introduced.

Questions & Answers

Q: Can you explain how the uncertainty principle can be defeated by measuring position and velocity simultaneously?

In order to measure the position of an object accurately, you need to hit it with a photon of a certain wavelength. However, this photon will change the velocity of the object. So, if you try to measure the velocity afterwards, you will discover that it has been altered. This means that you can't measure both the position and velocity of an object simultaneously without affecting the results.

Q: Can you explain the concept of linear vector spaces?

Linear vector spaces are sets of objects that can be multiplied by numbers and added together. In the case of complex vector spaces, the objects can be multiplied by complex numbers and added together. The simplest example of a complex vector space is the set of complex numbers themselves, which can be multiplied and added. Other examples include functions and column vectors.

Q: How do you represent complex numbers as vectors in a complex vector space?

Complex numbers can be represented as points on the xy-plane, with the real part as the x-coordinate and the imaginary part as the y-coordinate. Each complex number can also be represented as a ket vector in the complex vector space.

Q: What is the dual vector space and how is it related to complex conjugation?

The dual vector space is the complex conjugate vector space, where each vector in the original vector space is paired with its complex conjugate. In other words, for every ket vector, there is a corresponding bra vector, which is the complex conjugate of the ket vector.

Q: How does complex conjugation affect the multiplication of vectors in the dual vector space?

When multiplying a vector by a constant in the dual vector space, the image of that vector in the dual space is multiplied by the complex conjugate of the constant. This ensures that the inner product between two vectors in the dual space is properly calculated.

Q: What are the properties of the inner product between vectors?

The inner product is linear, meaning that it follows the distributive property and scales with constants. It is also conjugate symmetric, which means that the inner product of vector A with vector B is equal to the complex conjugate of the inner product of vector B with vector A. Additionally, the inner product of a vector with itself is real and positive.

Q: What is the specific inner product for functions?

The inner product of two functions phi and psi is given by the integral of the complex conjugate of phi multiplied by psi with respect to x. The interval of integration depends on the context.

Q: How is the inner product for column vectors defined?

The inner product of two column vectors A and B is given by the sum of the product of their corresponding entries, such as A1×B1 + A2×B2 + ... This follows the same pattern as the dot product of ordinary vectors in multiple dimensions.

Q: Can you provide an example of a complex vector space?

One example of a complex vector space is the space of complex functions, where the functions have a real part and an imaginary part. Another example is the space of column vectors with complex entries.

Takeaways

Linear vector spaces are sets of objects that can be multiplied by numbers and added together. Complex vector spaces are those in which the objects can be multiplied by complex numbers and added together. Complex numbers can be represented as points on the xy-plane and as ket vectors in the complex vector space. The dual vector space is formed by taking the complex conjugates of the vectors in the original vector space. The inner product between vectors is a linear operation that has properties such as linearity, conjugation, and positivity. The specific inner product for functions involves integrating the complex conjugate of one function multiplied by another function. Similarly, the inner product for column vectors involves summing the product of their corresponding entries.