As part of his preparations for the 1998 World Cup, Glenn Hoddle introduced the England team to Eileen Drewery, a faith healer. In her meetings with individual members of the squad, Drewery's habit was to begin the proceedings by placing her hands on the player's head. It was Ray Parlour, the Romford Pelé, who felt her touch and chirped: "Short back and sides, Eileen, if you please!"

If there were such a thing as your average English professional footballer, that would be how you might expect him to respond. Which makes it all the more interesting to hear the reaction of Steven Gerrard, the Liverpool and England captain, to the work of Dr Steve Peters, the sports psychiatrist who will be travelling to Brazil as a member of Roy Hodgson's party.

In his often guarded manner, Gerrard can display the familiar defensiveness with which footballers keep the outside world at bay. But two years of work with Peters, who was recommended to him by a club physio when he was struggling with a long-term groin injury, has been enough to banish any hint of flippant scepticism. And the success of their collaboration inspired Brendan Rodgers' decision to invite Peters to work with the whole Anfield squad.

"Steve Peters is not going to make the players run 100 metres any quicker, although he is a former sprinter himself, or do a Cruyff turn better or hit a 40-yard pass any more accurately," Gerrard had said. "But what I can guarantee is that if the players buy into it, he'll be able to help them with mental preparation and make them understand how the mind works, especially when you're going into pressure situations."

When I went to see Rodgers' first home game in charge – a 2-2 draw with Manchester City in August 2012 – it was obvious even at such an early stage that the young manager was bringing a new attitude to the team, symbolised in the decision to give the 17-year-old Raheem Sterling his full debut against the recently crowned champions. Their play was characterised by a sense of optimism so strong that it would not be dimmed by the reverses which inevitably came their way as the manager constructed a side capable of challenging for honours again.

That quality of optimism has been enhanced since Peters' arrival and is, I think, what has made this the most enjoyable Liverpool side in my experience, going back to the days of Ron Yeats and Ian St John. Not the best, but the one most likely to persuade a neutral to alter plans in order to stay in and see them on the television. And that, as much as their ascent to the top of the table, is what has made Anfield such a happy place in recent months.

Watching them dismiss Tottenham last weekend, two small incidents stuck in the mind. While Gerrard and Luis Suárez were sizing up a free‑kick outside the Spurs area, Jordan Henderson engaged Philippe Coutinho in a conversation of great seriousness and intensity: here were two players who, instead of taking a breather, were thinking hard about what they were doing. And when Henderson's own free-kick found its way into the net for Liverpool's fourth goal, Suárez – who could easily have claimed that it brushed him before crossing the line, thus giving him a 30th league goal this season – immediately turned and rushed towards the Englishman, pointing his finger and making it clear where the credit belonged.

It would be a great surprise to discover that Peters – whose presence has been crucial to the success of Britain's road and track cyclists, and whose ideas on mind management are laid out in a book called The Chimp Paradox – deserved no credit for the spread of this positive approach. Gerrard's endorsement, when asked about the psychiatrist's involvement with England on the eve of the recent friendly against Denmark, took his listeners into areas that footballers regard as off-limits to the media.

"I read his book and had one-on-one meetings with him," Gerrard said, "and I basically understand the different parts of the brain, how they work, when they function and why you think certain things – why you bite your children's heads off from time to time or blame Alex [his wife] for everything. I'm a lot more patient and I think I've improved as a person. He's also helped me with the game as well."

Gerrard considers himself to have a strong mentality, but he opted to continue the relationship after the injury problem had been resolved. "I don't want anyone to think he's transformed me as a player or a person, but I didn't know what was going on in my head until I saw him. And the way I look at it is that if it's going to help me add one, two or three per cent to what I've already got, why not use it?"

Most observers of Gerrard's career would agree that his performances this season, in a role at the base of midfield, display a new maturity. No longer overdoing it in an effort to prove his keenness to take responsibility, as he approaches his 34th birthday he has become a calmer player, and a more consistently effective one.

Hodgson, while warning that Dr Peters' presence is no guarantee of flawless penalty shoot-outs, expressed the hope that others will take note of what the psychiatrist has to offer. "All I can do is offer it up to the players," he said. "I'd expect them to say, especially the younger ones, 'If the guy can help a fellow like Steven Gerrard, perhaps he can help me, too.'"

In The Chimp Paradox, Peters teaches his readers how to distinguish between dreams and goals, and how to make one support the other. Liverpool go into the match at West Ham on Sunday on a run of eight consecutive league wins, knowing that victories in all their remaining half-dozen games would give them a first title in 24 years. The conditions – particularly the absence of European distractions and the effect of transition on Manchester United and Chelsea – might never be so propitious again.

If they manage to bring it off, Steve Peters will have another success to line up alongside the medals won by Chris Hoy, Victoria Pendleton and others who benefited from his ideas on how to achieve confidence, success and happiness. And if they don't, you can be pretty sure it won't be because they lacked the mental strength.