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The Physics of Quantum Mechanics

James Binney, David Skinner

Chapter 1

Introduction - all with Video Answers

Educators

Chapter Questions

01:03

Problem 1

What physical phenomenon requires us to work with probability amplitudes rather than just with probabilities, as in other fields of endeayour?

AG
Ankit Gupta
Numerade Educator
02:54

Problem 2

What properties cause complete sets of amplitudes to constitute the elements of a vector space?

Linda Winkler
Linda Winkler
Numerade Educator
01:00

Problem 3

$V^{\prime}$ is the dual space of the vector space $V$. For a mathematician, what objects comprise $V^{\prime \prime} ?$

Victor Salazar
Victor Salazar
Numerade Educator
00:59

Problem 4

In quantum mechanics, what objects are the members of the vector space $V ?$ Give an example for the case of quantum mechanics of a member of the dual space $V^{\prime}$ and explain how members of $V^{\prime}$ enable us to predict the outcomes of experiments.

Lottie Adams
Lottie Adams
Numerade Educator
04:00

Problem 5

Given that $|\psi\rangle=\mathrm{e}^{\mathrm{i} \pi / 5}|a\rangle+\mathrm{e}^{\mathrm{i} \pi / 4}|b\rangle$, express $\langle\psi|$ as a linear combination of $\langle a|$ and $\langle b|$.

M S
M S
Numerade Educator
01:54

Problem 6

What properties characterise the bra $\langle a|$ that is associated with the ket $|a\rangle ?$

Himanshu Garg
Himanshu Garg
Numerade Educator
02:03

Problem 7

An electron can be in one of two potential wells that are so close that it can 'tunnel' from one to the other (see $\S 5.2$ for a description of quantum-mechanical tunnelling). Its state vector can be written
$$
|\psi\rangle=a|A\rangle+b|B\rangle
$$
where $|A\rangle$ is the state of being in the first well and $|B\rangle$ is the state of being in the second well and all kets are correctly normalised. What is the probability of finding the particle in the first well given that: (a) $a=\mathrm{i} / 2 ;$ (b) $b=\mathrm{e}^{\mathrm{i} \pi} ;$ (c) $b=\frac{1}{\mathrm{v}}+\mathrm{i} / \sqrt{2} ?$

Hafiz Shahzaib
Hafiz Shahzaib
Numerade Educator
02:03

Problem 8

An electron can 'tunnel' between potential wells that form a chain, so its state vector can be written
$$
|\psi\rangle=\sum_{-\infty}^{\infty} a_{n}|n\rangle
$$
where $|n\rangle$ is the state of being in the $n^{\mathrm{th}}$ well, where $n$ increases from left to right. Let
$$
a_{n}=\frac{1}{\sqrt{2}}\left(\frac{-\mathrm{i}}{3}\right)^{|n| / 2} \mathrm{e}^{\mathrm{i} n \pi}
$$
a. What is the probability of finding the electron in the $n^{\text {th }}$ well?
b. What is the probability of finding the electron in well 0 or anywhere to the right of it?

Hafiz Shahzaib
Hafiz Shahzaib
Numerade Educator