Given that |ψ⟩=e^i π/ 5|a⟩+e^i π/ 4|b⟩, express ⟨ψ| as a linear combination of ⟨a| and ⟨b|.

Name: Problem 5, Chapter 1: The Physics of Quantum Mechanics
Uploaded: 2020-05-24T20:22:30-08:00
Duration: 4 min
Channel: M S
Description: Given that |ψ⟩=e^i π/ 5|a⟩+e^i π/ 4|b⟩, express ⟨ψ| as a linear combination of ⟨a| and ⟨b|.

Given that |ψ⟩=e^i π/ 5|a⟩+e^i π/ 4|b⟩, express ⟨ψ| as a...

Key Concepts

Complex Conjugation of Phase Factors

In quantum mechanics, phase factors such as exponential terms with imaginary arguments represent rotations in the complex plane. When taking the Hermitian conjugate of a state, these phase factors are complex-conjugated, meaning that any exponential term e^(i?) becomes e^(-i?). This process is critical to ensure physical quantities derived from the state, such as probabilities, are real and non-negative.

Linear Combination of States

Quantum states are often expressed as linear combinations of basis states. This principle reflects the superposition principle where any state in a Hilbert space can be represented as a sum of basis states, with corresponding coefficients that encapsulate information about the amplitude and phase. The ability to decompose states into basis components makes it easier to analyze and compute outcomes of quantum measurements.

Bra-Ket Notation

Bra-Ket notation is a standard formalism in quantum mechanics used to represent state vectors and their duals. In this system, kets (e.g., |??) denote vectors in a Hilbert space, while bras (e.g., ??|) represent their corresponding dual vectors (linear functionals). This notation facilitates inner product calculations and the elegant representation of quantum states and their transformations.

Hermitian Conjugate (Adjoint) Operation

The Hermitian conjugate, or adjoint, of an operator or state vector involves taking the complex conjugate of its coefficients and transposing it. When applied to a quantum state written as a linear combination of kets, the Hermitian conjugate converts the state into its corresponding bra by replacing each ket with its bra and applying complex conjugation to the coefficients. This operation ensures that inner products yield real values, as required in quantum mechanics.

Transcript

00:01 Suppose we're given that wave functions psi 1 and psi 2 are solutions to the schrodinger equation copied here.

00:11 We want to show that if this is the case, then some linear combination of these solutions, also there are solutions with the same energy e.

00:23 Then we want to show that this linear combination of those wave functions with arbitrary constants a and b is also a solution, also with energy e.

00:32 And so what we'll do is we'll start by plugging in.

00:37 What we'll do is we'll show that if we use this new wave function, on the left side of the equation, we will get the right side of the equation with the same energy.

00:46 So we'll go ahead and start.

00:48 We know that this is true for psi 1 and side 2, d x squared.

00:55 And so for psi, we're plugging in our linear combination of these solutions.

01:01 Then we've got u.

01:02 We might write u of x.

01:05 Then here's our equation.

01:10 Okay.

01:12 So we want to show that this is equal to e, a, psi 1 plus b, side 2.

01:18 And so to do that, we're going to rearrange this and use the fact that psi 1 and psi 2 are both solutions.

01:27 And so we'll go ahead and rearrange this and group the psi 1 terms and group the si 2 terms.

01:34 So h bar squared over 2m d squared d x squared of a a...